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3D Gyrokinetic Plasma Turbulence Simulations
Impact of the temperature ratio on turbulent impurity transport in Wendelstein 7-X
Th. Wegner, J.A. Alcusón, B. Geiger, A. v. Stechow, P. Xanthopoulos, C. Angioni, M.N.A. Beurskens, L.-G. Böttger, S.A. Bozhenkov, K.J. Brunner, R. Burhenn, B. Buttenschön, H. Damm, E. Edlund, O.P. Ford, G. Fuchert, O. Grulke, Z. Huang, J. Knauer, F. Kunkel, A. Langenberg, N.A. Pablant, E. Pasch, K. Rahbarnia, J. Schilling, H. Thomsen, L. Vanó and W7-X team
First experimental observations in the Wendelstein 7-X stellarator indicate that the impurity confinement can be explained by turbulent processes. In particular, plasma discharges with increased ion to electron temperature ratio are accompanied by reduced electron density fluctuation amplitudes and anomalous impurity diffusion, suggesting a lower turbulent transport. Employing gyro-kinetic numerical simulations, we argue that the temperature ratio plays a key role for reducing the ion temperature gradient instability in Wendelstein 7-X, leading to an enhanced impurity confinement.
Turbulence Mechanisms of Enhanced Performance Stellarator Plasmas
P. Xanthopoulos, S. A. Bozhenkov, M. N. Beurskens, H. M. Smith, G. G. Plunk, P. Helander, C. D. Beidler, J. A. Alcusón, A. Alonso, A. Dinklage, O. Ford, G. Fuchert, J. Geiger, J. H. E. Proll, M. J. Pueschel, Y. Turkin, F. Warmer and the W7-X Team
We theoretically assess two mechanisms thought to be responsible for the enhanced performance observed in plasma discharges of the Wendelstein 7-X stellarator experiment fueled by pellet injection. The effects of the ambipolar radial electric field and the electron density peaking on the turbulent ion heat transport are separately evaluated using large-scale gyrokinetic simulations. The essential role of the stellarator magnetic geometry is demonstrated, by comparison with a tokamak.
High-performance plasmas after pellet injections in Wendelstein 7-X
S. A. Bozhenkov, Y. Kazakov, O. P. Ford, M. N. A. Beurskens, J. Alcusón, J. A. Alonso, J. Baldzuhn, C. Brandt, K. J. Brunner, H. Damm, G. Fuchert, J. Geiger, O. Grulke, M. Hirsch, U. Höfel, Z. Huang, J. Knauer, M. Krychowiak, A. Langenberg, H. P. Laqua, S. Lazerson, N. B. Marushchenko, D. Moseev, M. Otte, N. Pablant, E. Pasch, A. Pavone, J. H. E. Proll, K. Rahbarnia, E. R. Scott, H. M. Smith, T. Stange, A. von Stechow, H. Thomsen, Yu. Turkin, G. Wurden, P. Xanthopoulos, D. Zhang, R. C. Wolf and the W7-X team
A significant improvement of plasma parameters in the optimized stellarator W7-X is found after injections of frozen hydrogen pellets. The ion temperature in the post-pellet phase exceeds 3 keV with 5 MW of electron heating and the global energy confinement time surpasses the empirical ISS04-scaling. The plasma parameters realized in such experiments are significantly above those in comparable gas-fueled discharges. In this paper, we present details of these pellet experiments and discuss the main plasma properties during the enhanced confinement phases. Local power balance is applied to show that the heat transport in post-pellet phases is close to the neoclassical level for the ion channel and is about a factor of two above that level for the combined losses. In comparable gas-fueled discharges, the heat transport is by about ten times larger than the neoclassical level, and thus is largely anomalous. It is further observed that the improvement in the transport is related to the peaked density profiles that lead to a stabilization of the ion-scale turbulence.
Global gyrokinetic simulations of ITG turbulence in the magnetic configuration space of the Wendelstein 7-X stellarator
A. Bañón Navarro, G. Merlo, G. G. Plunk, P. Xanthopoulos, A. von Stechow, A. Di Siena, M. Maurer, F. Hindenlang, F. Wilms and F. Jenko
We study the effect of turbulent transport in different magnetic configurations of the Weldenstein 7-X stellarator. In particular, we performed direct numerical simulations with the global gyrokinetic code GENE-3D, modeling the behavior of Ion Temperature Gradient turbulence in the Standard, High-Mirror and Low-Mirror configurations of W7-X. We found that the Low-Mirror configuration produces more transport than both the High-Mirror and the Standard configurations. By comparison with radially local simulations, we have demonstrated the importance of performing global non-linear simulations to predict the turbulent fluxes quantitatively.
Geometric stabilization of the electrostatic ion-temperature-gradient driven instability. II. Non-axisymmetric systems
A non-perturbative analysis for the of study non-axisymmetric (3D) effects on the linear ion-temperature-gradient driven mode is introduced. Perturbations and equilibria are considered to be global on the flux surface, yet radially local. The analysis is valid for systems arbitrarily far from axisymmetry. It is found that finite Larmor radius effects can suppress the global (on the surface) instability, in analogy with the local analysis, but shift its poloidal location from the position of the greatest local instability. Fourier spectra of the instability whose width grow for increasingly non-axisymmetric systems are predicted. Results are in qualitative agreement with numerical global (on the surface) gyrokinetic simulations.
Suppression of electrostatic micro-instabilities in maximum-J stellarators
J. A. Alcusón, P. Xanthopoulos, G. G. Plunk, P. Helander, F. Wilms, Y. Turkin, A. von Stechow and O. Grulke
We demonstrate favorable stability properties of maximum-J stellarators, exemplified by the Wendelstein 7-X (W7-X) device, in scenarios with low plasma beta. A large number of electrostatic linear gyrokinetic simulations are conducted to scan the relevant parameter space for different configurations, resulting in stability maps that account for the key micro-instabilities thought to drive turbulent transport. These maps exhibit a ``stability valley'' in the region where the normalized ion temperature gradient is roughly equal to the normalized density gradient. In this valley, the electrostatic instabilities are partly suppressed thanks to the maximum-J property of the W7-X field. This property varies across different W7-X configurations, and this measurable difference is demonstrated to affect the size of the stability valley. Finally, the impact of the isotope effect and collisions on the valley is examined.
A comparison of turbulent transport in a quasi-helical and a quasi-axisymmetric stellarator
Ion-temperature-gradient-driven (ITG) turbulence is compared for two quasi-symmetric (QS) stellarator configurations to determine the relationship between linear growth rates and nonlinear heat fluxes. We focus on the quasi-helically symmetric (QHS) stellarator HSX and the quasi-axisymmetric (QAS) stellarator NCSX. In normalized units, HSX exhibits higher growth rates than NCSX, while heat fluxes in gyro-Bohm units are lower in HSX. These results hold for simulations made with both adiabatic and kinetic electrons. The results show that HSX has a larger number of subdominant modes than NCSX and that eigenmodes are more spatially extended in HSX. We conclude that the consideration of nonlinear physics is necessary to accurately assess the heat flux due to ITG turbulence when comparing QS stellarator equilibria.
Electron temperature gradient driven instabilities in helical reversed-field pinch plasmas
We describe the occurrence of electron temperature gradient driven micro-instabilities in the helical states of RFX-mod reversed-field pinch plasmas. These plasmas are usually characterized by the presence of large, radially localized electron temperature gradients. Using realistic geometry and profiles, micro-tearing modes and electron temperature gradient modes turn out to regularly co-exist in the region of the temperature barrier. In the paper, we discuss the main features of such instabilities, especially focusing on the distinctive aspects of the helical geometry with respect to the toroidal symmetry.
Stellarators resist turbulent transport on the electron Larmor scale
G. G. Plunk, P. Xanthopoulos, G. M. Weir, S. A. Bozhenkov, A. Dinklage, G. Fuchert, J. Geiger,
M. Hirsch, U. Hoefel, M. Jakubowski, A. Langenberg, N. Pablant, E. Pasch, T. Stange, D. Zhang
Electron temperature gradient (ETG)-driven turbulence, despite its ultrafine scale, is thought to drive significant thermal losses in magnetic fusion devices—but what role does it play in stellarators? The first numerical simulations of ETG turbulence for the Wendelstein 7-X stellarator, together with power balance analysis from its initial experimental operation phase, suggest that the associated transport should be negligible compared to other channels. The effect, we argue, originates essentially from the geometric constraint of multiple field periods, a generic feature of stellarators.
Ion temperature gradient turbulence modification in quasi-axisymmetry
S. A. Lazerson, P. Xanthopoulos, H. Mynick and D. Gates
The large flexibility of the proposed QUASAR facility [Gates et al., Nucl. Fusion 57, 126064 (2017)] is leveraged in order to explore the effect of magnetic shear on adiabatic Ion Temperature Gradient (ITG) turbulence. The QUASAR facility is a reimagining of the National Compact Stellarator Experiment utilizing and expanding upon the already constructed coil set. Recent work using fixed boundary optimization of the LI383 equilibrium (upon which QUASAR is based) has suggested possible improvements to ITG turbulence [Mynick et al., Plasma Phys. Controlled Fusion 56, 094001 (2014)]. In this work, a different approach is taken, wherein a series of self-consistent free boundary VMEC equilibria are developed for QUASAR. These equilibria assume temperature and density profiles consistent with 2% beta and ohmic current drive. In each configuration, the toroidal field coils are energized to different values and the STELLOPT code is used to vary the modular coil current and net toroidal current.
Linear electrostatic gyrokinetics for electron–positron plasma
D. Kennedy, A. Mischchenko, P. Xanthopoulos and P. Helander
Gyrokinetic stability of plasmas in different magnetic geometries is studied numerically using the GENE code. We examine the stability of plasmas, varying the mass ratio between the positive and negative charge carriers, from conventional hydrogen plasmas through to electron–positron plasmas. Stability is studied for prescribed temperature and density gradients in different magnetic geometries: (i) An axisymmetric, circular flux surface, low 𝛽 (tokamak) configuration. (ii) A non-axisymmetric quasi-isodynamic (optimised stellarator) configuration using the geometry of the stellarator Wendelstein 7-X. We also present the analytic theory of trapped particle modes in electron–positron plasmas. We found similar behavior of the growth rate and real frequency compared to previous studies on the tokamak case.
Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas
A. Zocco, P. Xanthopoulos, H. Doerk, J. W. Connor and P. Helander
The threshold for the resonant destabilisation of ion-temperature-gradient (ITG) driven instabilities that render the modes ubiquitous in both tokamaks and stellarators is investigated. We discover remarkably similar results for both confinement concepts if care is taken in the analysis of the effect of the global shear. We revisit, analytically and by means of gyrokinetic simulations, accepted tokamak results and discover inadequacies of some aspects of their theoretical interpretation. In particular, for standard tokamak configurations, we find that global shear effects on the critical gradient cannot be attributed to the wave–particle resonance destabilising mechanism of Hahm & Tang (Phys. Plasmas, vol. 1, 1989, pp. 1185–1192), but are consistent with a stabilising contribution predicted by Biglari et al. (Phys. Plasmas, vol. 1, 1989, pp. 109–118).
Strongly driven surface-global kinetic ballooning modes in general toroidal geometry
Kinetic ballooning modes in magnetically confined toroidal plasmas are investigated putting emphasis on specific stellarator features. In particular, we propose a Mercier criterion which is purposely designed to allow for direct comparison with local flux-tube gyrokinetics simulations. We investigate the influence on the marginal frequency of the mode of a magnetic curvature which is inhomogeneous on the magnetic flux surface due to the fieldline-label dependence. This is a typical (surface) global effect present in non-axisymmetry. Finally, we propose an artificial equilibrium model that explicitly retains the fieldline-label dependence in the magnetic drift, and analyse the stability of the system by introducing a representation of the perturbations similar to the flux-bundle model of Sugama et al. (Plasma Fusion Res., vol. 7, 2012, 2403094). The coupling of flux bundles is shown to have a stabilising effect on the most unstable local flux-tube mode.
Kinetic ballooning modes in tokamaks and stellarators
K. Aleynikova, A. Zocco, P. Xanthopoulos, P. Helander and C. Nührenberg
Kinetic ballooning modes (KBMs) are investigated by means of linear electromagnetic gyrokinetic (GK) simulations in the stellarator Wendelstein 7-X (W7-X), for high-β plasmas, where β is the ratio of thermal to magnetic plasma pressure. The analysis shows suppression of ion-temperature-gradient (ITG) and trapped particle modes (TEM) by finite-β effects and destabilization of KBMs at high β. The results are compared with a generic tokamak case.
The parallel boundary condition for turbulence simulations in low magnetic shear devices
M. F. Martin, M. Landreman, P. Xanthopoulos, N. R. Mandell and W. Dorland
Flux tube simulations of plasma turbulence in stellarators and tokamaks typically employ coordinates which are aligned with the magnetic field lines. Anisotropic turbulent fluctuations can be represented in such field-aligned coordinates very efficiently, but the resulting non-trivial boundary conditions involve all three spatial directions, and must be handled with care. The standard 'twist-and-shift' formulation of the boundary conditions (Beer et al, 1995 Phys. Plasmas 2, 2687) was derived assuming axisymmetry and is widely used because it is efficient, as long as the global magnetic shear is not too small. A generalization of this formulation is presented, appropriate for studies of non-axisymmetric, stellarator symmetric configurations, as well as for axisymmetric configurations with small global shear. The key idea is to replace the 'twist' of the standard approach (which accounts only for global shear) with the integrated local shear.
First steps towards modeling of ion-driven turbulence in Wendelstein 7-X
F. Warmer, P. Xanthopoulos, J. H. E. Proll, C. D. Beidler, Y. Turkin and R. C. Wolf
Due to foreseen improvement of neoclassical confinement in optimised stellarators—like the newly commissioned Wendelstein 7-X (W7-X) experiment in Greifswald, Germany—it is expected that turbulence will significantly contribute to the heat and particle transport, thus posing a limit to the performance of such devices. In order to develop discharge scenarios, it is thus necessary to develop a model which could reliably capture the basic characteristics of turbulence and try to predict the levels thereof. The outcome will not only be affordable, using only a fraction of the computational cost which is normally required for repetitive direct turbulence simulations, but would also highlight important physics. In this model, we seek to describe the ion heat flux caused by ion temperature gradient (ITG) micro-turbulence, which, in certain heating scenarios, can be a strong source of free energy.
Recent advances in stellarator optimization
D. A. Gates, A. H. Boozer, T. Brown, J. Breslau, D. Curreli, M. Landreman, S. A. Lazerson, J. Lore, H. Mynick, G. H. Neilson, N. Pomphrey, P. Xanthopoulos and A. Zolfaghari
Nuclear Fusion 58, 016017 (2018)
Computational optimization has revolutionized the field of stellarator design. To date, optimizations have focused primarily on optimization of neoclassical confinement and ideal MHD stability, although limited optimization of other parameters has also been performed. The purpose of this paper is to outline a select set of new concepts for stellarator optimization that, when taken as a group, present a significant step forward in the stellarator concept. One of the criticisms that has been leveled at existing methods of design is the complexity of the resultant field coils. Recently, a new coil optimization code—COILOPT++, which uses a spline instead of a Fourier representation of the coils,—was written and included in the STELLOPT suite of codes. The advantage of this method is that it allows the addition of real space constraints on the locations of the coils. The code has been tested by generating coil designs for optimized quasi-axisymmetric stellarator plasma configurations of different aspect ratios. As an initial exercise, a constraint that the windings be vertical was placed on large major radius half of the non-planar coils. Further constraints were also imposed that guaranteed that sector blanket modules could be removed from between the coils, enabling a sector maintenance scheme. Results of this exercise will be presented. New ideas on methods for the optimization of turbulent transport have garnered much attention since these methods have led to design concepts that are calculated to have reduced turbulent heat loss. We have explored possibilities for generating an experimental database to test whether the reduction in transport that is predicted is consistent with experimental observations. To this end, a series of equilibria that can be made in the now latent QUASAR experiment have been identified that will test the predicted transport scalings. Fast particle confinement studies aimed at developing a generalized optimization algorithm are also discussed. A new algorithm developed for the design of the scraper element on W7-X is presented along with ideas for automating the optimization approach.
Overview of the RFX-mod fusion science activity
M. Zuin, S. Dal Bello, L. Marrelli, M. E. Puiatti, P. Agostinetti, M. Agostini, V. Antoni, F. Auriemma, M. Barbisan, T. Barbui et al.
This paper reports the main recent results of the RFX-mod fusion science activity. The RFX-mod device is characterized by a unique flexibility in terms of accessible magnetic configurations. Axisymmetric and helically shaped reversed-field pinch equilibria have been studied, along with tokamak plasmas in a wide range of q(a) regimes (spanning from 4 down to 1.2 values). The full range of magnetic configurations in between the two, the so-called ultra-low q ones, has been explored, with the aim of studying specific physical issues common to all equilibria, such as, for example, the density limit phenomenon. The powerful RFX-mod feedback control system has been exploited for MHD control, which allowed us to extend the range of experimental parameters, as well as to induce specific magnetic perturbations for the study of 3D effects. In particular, transport, edge and isotope effects in 3D equilibria have been investigated, along with runaway mitigations through induced magnetic perturbations. The first transitions to an improved confinement scenario in circular and D-shaped tokamak plasmas have been obtained thanks to an active modification of the edge electric field through a polarized electrode. The experiments are supported by intense modeling with 3D MHD, gyrokinetic, guiding center and transport codes. Proposed modifications to the RFX-mod device, which will enable further contributions to the solution of key issues in the roadmap to ITER and DEMO, are also briefly presented.
The effect of transient density profile shaping on transport in large stellarators and heliotrons
A. Dinklage, R. Sakamoto, M. Yokoyama, K. Ida, J. Baldzuhn, C. D. Beidler, S. Cats, K. J. Mc Carthy, J. Geiger, M. Kobayashi, H. Maaßberg, S. Morita, G. Motojima, M. Nakata, M. Nunami, N. Pablant, K. Ogawa, J. H. E. Proll, S. Satake, K. Tanaka, F. Warmer, R. C. Wolf, P. Xanthopoulos, H. Yamada, R. Yasuhara and M. Yoshinuma
Transport studies of pellet fueling experiments on LHD are reported. Spatio-temporal evolutions after pellet injection into LHD discharges show cases where central density increases on the time scale of particle transport processes. Both the temperature gradient and the density gradient change during the density relaxation, the latter even in sign. The resulting thermodynamic forces influence radial electric fields—both as a driving term but also by, e.g. affecting the Er dependence of ion transport. Magnetic fluctuations have been found to be induced by pellet injection but die out with the relaxation of the pressure profile.
Distinct Turbulence Saturation Regimes in Stellarators
In the complex 3D magnetic fields of stellarators, ion-temperature-gradient turbulence is shown to have two distinct saturation regimes, as revealed by petascale numerical simulations and explained by a simple turbulence theory. The first regime is marked by strong zonal flows and matches previous observations in tokamaks. The newly observed second regime, in contrast, exhibits small-scale quasi-two-dimensional turbulence, negligible zonal flows, and, surprisingly, a weaker heat flux scaling. Our findings suggest that key details of the magnetic geometry control turbulence in stellarators.
Quasi-symmetry and the nature of radial turbulent transport in quasi-poloidal stellarators
J. A. Alcusón, J. M. Reynolds-Barredo, A. Bustos, R. Sanchez, V. Tribaldos, P. Xanthopoulos, T. Goerler and D. E. Newman
Quasi-symmetric configurations have a better neoclassical confinement compared to that of standard stellarators. The reduction of the neoclassical viscosity along the direction of quasi-symmetry should facilitate the self-generation of zonal flows and, consequently, the mitigation of turbulent fluctuations and the ensuing radial transport. Therefore, it is expected that quasi-symmetries should also result in better confinement properties regarding radial turbulent transport. In this paper we show that, at least for quasi-poloidal configurations, the influence of quasi-symmetry on radial transport exceeds the expected reduction of fluctuation levels and associated effective transport coefficients, and that the intimate nature of transport itself is affected. In particular, radial turbulent transport becomes increas-ingly subdiffusive as the degree of quasi-symmetry becomes larger. This behavior is somewhat remi-niscent of what has been previously reported in tokamaks with strong radially sheared zonal flows.
Geometric stabilization of the electrostatic ion-temperature-gradient driven instability. I. Nearly axisymmetric systems
A. Zocco, G. G. Plunk, P. Xanthopoulos and P. Helander
The effects of a non-axisymmetric (3D) equilibrium magnetic field on the linear ion-temperature-gradient (ITG) driven mode are investigated. We consider the strongly driven, toroidal branch of the instability in a global (on the magnetic surface) setting. Previous studies have focused on particular features of non-axisymmetric systems, such as strong local shear or magnetic ripple, that introduce inhomogeneity in the coordinate along the magnetic field. In contrast, here we include non-axisymmetry explicitly via the dependence of the magnetic drift on the field line label α, i.e., across the magnetic field, but within the magnetic flux surface. We consider the limit where this variation occurs on a scale much larger than that of the ITG mode, and also the case where these scales are similar. Close to axisymmetry, we find that an averaging effect of the magnetic drift on the flux surface causes global (on the surface) stabilization, as compared to the most unstable local mode. In the absence of scale separation, we find destabilization is also possible, but only if a particular resonance occurs between the magnetic drift and the mode, and finite Larmor radius effects are neglected. We discuss the relative importance of surface global effects and known radially global effects.
Intrinsic Turbulence Stabilization in a Stellarator
P. Xanthopoulos, G. G. Plunk, A. Zocco, and P. Helander
The magnetic surfaces of modern stellarators are characterized by complex, carefully optimized shaping and exhibit locally compressed regions of strong turbulence drive. Massively parallel computer simulations of plasma turbulence reveal, however, that stellarators also possess two intrinsic mechanisms to mitigate the effect of this drive. In the regime where the length scale of the turbulence is very small compared to the equilibrium scale set by the variation of the magnetic field, the strongest fluctuations form narrow bandlike structures on the magnetic surfaces. Thanks to this localization, the average transport through the surface is significantly smaller than that predicted at locations of peak turbulence. This feature results in a numerically observed upshift of the onset of turbulence on the surface towards higher ion temperature gradients as compared with the prediction from the most unstable regions. In a second regime lacking scale separation, the localization is lost and the fluctuations spread out on the magnetic surface. Nonetheless, stabilization persists through the suppression of the large eddies (relative to the equilibrium scale), leading to a reduced stiffness for the heat flux dependence on the ion temperature gradient. These fundamental differences with tokamak turbulence are exemplified for the QUASAR stellarator [G. H. Neilson et al., IEEE Trans. Plasma Sci. 42, 489 (2014)].
TEM turbulence optimisation in stellarators
J. H.E Proll, H. E Mynick, P. Xanthopoulos, S. A. Lazerson and B. J. Faber
With the advent of neoclassically optimised stellarators, optimising stellarators for turbulent transport is an important next step. The reduction of ion-temperature-gradient-driven turbulence has been achieved via shaping of the magnetic field, and the reduction of trapped-electron mode (TEM) turbulence is addressed in the present paper. Recent analytical and numerical findings suggest TEMs are stabilised when a large fraction of trapped particles experiences favorable bounce-averaged curvature. This is the case for example in Wendelstein 7-X (Beidler et al 1990 Fusion Technol. 17, 148) and other Helias-type stellarators. Using this knowledge, a proxy function was designed to estimate the TEM dynamics, allowing optimal configurations for TEM stability to be determined with the STELLOPT (Spong et al 2001 Nucl. Fusion 41, 711) code without extensive turbulence simulations. A first proof-of-principle optimised equilibrium stemming from the TEM-dominated stellarator experiment HSX (Anderson et al 1995 Fusion Technol. 27, 273) is presented for which a reduction of the linear growth rates is achieved over a broad range of the operational parameter space. As an important consequence of this property, the turbulent heat flux levels are reduced compared with the initial configuration.
Gyrokinetic studies of trapped electron mode turbulence in the Helically Symmetric eXperiment stellarator
B. J. Faber, M. J. Pueschel, J. H. E. Proll, P. Xanthopoulos, P. W. Terry, C. C. Hegna, G. M. Weir, K. M. Likin and J. N. Talmadge
Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be over-whelmed at higher density gradients by the long-wavelength coherent structures.
Ion temperature gradient turbulence in helical and axisymmetric RFP plasmas
I. Predebon and P. Xanthopoulos
Physics of Plasmas 22, 052308 (2015)
Turbulence induced by the ion temperature gradient (ITG) is investigated in the helical and axisymmetric plasma states of a reversed field pinch device by means of gyrokinetic calculations. The two magnetic configurations are systematically compared, both linearly and nonlinearly, in order to evaluate the impact of the geometry on the instability and its ensuing transport, as well as on the production of zonal flows. Despite its enhanced confinement, the high-current helical state demonstrates a lower ITG stability threshold compared to the axisymmetric state, and ITG turbulence is expected to become an important contributor to the total heat transport.
Advances in stellarator gyrokinetics
P. Helander, T. Bird, F. Jenko, R. Kleiber, G. G. Plunk, J. H. E. Proll, J. Riemann and P. Xanthopoulos
Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ∗ even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.
Overview of the RFX-mod contribution to the international Fusion Science Program
The RFX-mod device is operated both as a reversed field pinch (RFP), where advanced regimes featuring helical shape develop, and as a tokamak. Due to its flexibility, RFX-mod is contributing to the solution of key issues in the roadmap to ITER and DEMO, including MHD instability control, internal transport barriers, edge transport and turbulence, isotopic effect, high density limit and three-dimensional (3D) non-linear MHD modelling. This paper reports recent advancements in the understanding of the self-organized helical states, featuring a strong electron transport barrier, in the RFP configuration; the physical mechanism driving the residual transport at the barrier has been investigated. Following the first experiments with deuterium as the filling gas, new results concerning the isotope effect in the RFP are discussed. Studies on the high density limit show that in the RFP it is related to a toroidal particle accumulation due to the onset of a convective cell. In the tokamak configuration, q(a) regimes down to q(a)=1.2 have been pioneered, with (2,1) tearing mode (TM) mitigated and (2,1) resistive wall mode (RWM) stabilized: the control of such modes can be obtained both by poloidal and radial sensors. Progress has been made in the avoidance of disruptions due to the (2,1) TM by applying q(a) control, and on the general issue of error field control. The effect of externally applied 3D fields on plasma flow and edge turbulence, sawtooth control and runaway electron decorrelation has been analysed. The experimental program is supported by substantial theoretical activity: 3D non-linear visco-resistive MHD and non-local transport modelling have been advanced; RWMs have been studied by a toroidal MHD kinetic hybrid stability code.
In order to study and design next-step fusion devices such as DEMO, comprehensive systems codes are commonly employed. In this work HELIAS-specific models are proposed which are designed to be compatible with systems codes. The subsequently developed models include: a geometry model based on Fourier coefficients which can represent the complex 3-D plasma shape, a basic island divertor model which assumes diffusive cross-field transport and high radiation at the X-point, and a coil model which combines scaling aspects based on the HELIAS 5-B reactor design in combination with analytic inductance and field calculations. In addition, stellarator-specific plasma transport is discussed. A strategy is proposed which employs a predictive confinement time scaling derived from 1-D neoclassical and 3-D turbulence simulations. This paper reports on the progress of the development of the stellarator-specific models while an implementation and verification study within an existing systems code will be presented in a separate work. This approach is investigated to ultimately allow one to conduct stellarator system studies, develop design points of HELIAS burning plasma devices, and to facilitate a direct comparison between tokamak and stellarator DEMO and power plant designs.
Implementation and verification of a HELIAS module for the systems code PROCESS
Fusion Engineering and Design 98-99, 2227-2230 (2015)
In order to study design points of next-step fusion devices such as DEMO, comprehensive systems codes are commonly employed. The code package PROCESS is such a tool, widely used for tokamak systems studies. In this work, the implementation and verification of a HELIAS module into PROCESS is addressed. These HELIAS models include: a plasma geometry model based on Fourier coefficients, a basic island divertor model, as well as a coil model which combines scaling aspects based on the Helias 5-B reactor design in combination with analytic inductance and field calculations. The models are verified firstly with respect to W7-X. Secondly, the generality of the models is used to represent the tokamak which is compared against the original tokamak PROCESS models using a DEMO design as reference case. Both approaches show very good agreement.
Statistical analysis and modeling of intermittent transport events in thetokamak scrape-off layer
J. Anderson, F. D. Halpern, P. Xanthopoulos, P. Ricci and I. Furno
The turbulence observed in the scrape-off-layer of a tokamak is often characterized by intermittent events of bursty nature, a feature which raises concerns about the prediction of heat loads on the physical boundaries of the device. It appears thus necessary to delve into the statistical properties of turbulent physical fields such as density, electrostatic potential, and temperature, focusing on the mathematical expression of tails of the probability distribution functions. The method followed here is to generate statistical information from time-traces of the plasma density stemming from Braginskii-type fluid simulations and check this against a first-principles theoretical model. The analysis of the numerical simulations indicates that the probability distribution function of the intermittent process contains strong exponential tails, as predicted by the analytical theory.
Controlling Turbulence in Present and Future Stellarators
P. Xanthopoulos, H. E. Mynick, P. Helander, Y. Turkin, G. G. Plunk, F. Jenko, T. Görler, D. Told, T. Bird, and J. H. E. Proll
Turbulence is widely expected to limit the confinement and, thus, the overall performance of modern neoclassically optimized stellarators. We employ novel petaflop-scale gyrokinetic simulations to predict the distribution of turbulence fluctuations and the related transport scaling on entire stellarator magnetic surfaces and reveal striking differences to tokamaks. Using a stochastic global-search optimization method, we derive the first turbulence-optimized stellarator configuration stemming from an existing quasi-omnigenous design.
Turbulent optimization of toroidal configurations
H. Mynick, P. Xanthopoulos, B. Faber, M. Lucia, M. Rorvig and J. N. Talmadge
Recent progress in ‘turbulent optimization’ of toroidal configurations is described, using a method recently developed for evolving such configurations to ones having reduced turbulent transport. The method uses the GENE gyrokinetic code to compute the radial heat flux Qgk, and the STELLOPT optimization code with a theory-based ‘proxy’ figure of merit Qpr to stand in for Qgk for computational speed. Improved expressions for Qpr have been developed, involving further geometric quantities beyond those in the original proxy, which can also be used as ‘control knobs’ to reduce Qgk. Use of a global search algorithm has led to the discovery of turbulent-optimized configurations not found by the standard, local algorithm usually employed, as has use of a mapping capability which STELLOPT has been extended to provide, of figures of merit over the search space.
Collisionless microinstabilities in stellarators. III. The ion-temperature-gradient mode
G. G. Plunk, P. Helander, P. Xanthopoulos and J. W. Connor
We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. A second model investigates the long-wavelength limit for the case of negligible global magnetic shear. The analytic calculation reveals that the effect of the curvature drive enters at second order in the drift frequency, confirming conventional wisdom that the ITG mode is slab-like at long wavelengths. Using flux tube simulations of a zero-shear W7-X configuration, we observe a close relationship to an axisymmetric configuration at a similar parameter point. It is concluded that scale lengths of the equilibrium gradients constitute a good parameter space to characterize the ITG mode. Thus, to optimize the magnetic geometry for ITG mode stability, it may be fruitful to focus on local parameters, such as the magnitude of bad curvature, connection length, and local shear at locations of bad curvature (where the ITG mode amplitude peaks).
Technical challenges in the construction of thesteady-state stellarator Wendelstein 7-X
The next step in the Wendelstein stellarator line is the large superconducting device Wendelstein 7-X, currently under construction in Greifswald, Germany. Steady-state operation is an intrinsic feature of stellarators, and one key element of the Wendelstein 7-X mission is to demonstrate steady-state operation under plasma conditions relevant for a fusion power plant. Steady-state operation of a fusion device, on the one hand, requires the implementation of special technologies, giving rise to technical challenges during the design, fabrication and assembly of such a device. On the other hand, also the physics development of steady-state operation at high plasma performance poses a challenge and careful preparation. The electron cyclotron resonance heating system, diagnostics, experiment control and data acquisition are prepared for plasma operation lasting 30 min. This requires many new technological approaches for plasma heating and diagnostics as well as new concepts for experiment control and data acquisition.
Gyrokinetic studies of the effect of β on drift-wave stability in the National Compact Stellarator Experiment
J. A. Baumgaertel, G. W. Hammett, D. R. Mikkelsen, M. Nunami and P. Xanthopoulos
The gyrokinetic turbulence code GS2 was used to investigate the effects of plasma β on linear, collisionless ion temperature gradient (ITG) modes and trapped electron modes (TEM) in National Compact Stellarator Experiment (NCSX) geometry. Plasma β affects stability in two ways: through the equilibrium and through magnetic fluctuations. The first was studied here by comparing ITG and TEM stability in two NCSX equilibria of differing β values, revealing that the high β equilibrium was marginally more stable than the low β equilibrium in the adiabatic-electron ITG mode case. However, the high β case had a lower kinetic-electron ITG mode critical gradient. Electrostatic and electromagnetic ITG and TEM mode growth rate dependencies on temperature gradient and density gradient were qualitatively similar. The second β effect is demonstrated via electromagnetic ITG growth rates’ dependency on GS2’s β input parameter. A linear benchmark with gyrokinetic codes GENE and GKV-X is also presented.
Gyrokinetic TEM stability calculations for quasi-isodynamic stellarators
J.H.E. Proll, P. Helander, P. Xanthopoulos and J. W. Connor
Recent theoretical findings suggest that in perfectly quasi-isodynamic stellarators, the trapped-particle instability as well as the ordinary electron-density-gradient-driven trapped-electron mode are stable in the electrostatic and collisionless approximation. In these configurations contours of constant magnetic field strength B are poloidally closed, the second adiabatic invariant J is constant on flux surfaces and peaks in the centre. It follows that the diamagnetic drift frequency ω∗a and the bounce-averaged magnetic drift frequency ωda are in opposite directions, ω∗a·ωda<0, everywhere on the flux surface. This is the signature of average “good curvature” for trapped particles and, thanks to this property, particles that bounce faster than the frequency of any unstable mode must draw energy from it near marginal stability. Consequently, the point of marginal stability cannot exist for the collisionless trapped-particle mode, and hence this mode will be absent. By a similar argument, the ordinary trapped-electron mode is also stable. Because perfect quasi-isodynamicity can never be reached, it is necessary to test configurations approaching quasi-isodynamicity numerically in order to probe their resilience against trapped-particle modes. Progress has been made to extend gyrokinetic simulations to stellarator geometries, enabling us to perform the first linear gyrokinetic simulations of density-gradient-driven modes in stellarator configurations approaching quasi-isodynamicity, such as Wendelstein 7-X and more recently found configurations. These simulations appear to confirm the analytical predictions.
Stellarator and tokamak plasmas: a comparison
P. Helander, C. D. Beidler, T. M. Bird, M. Drevlak, Y. Feng, R. Hatzky, F. Jenko, R. Kleiber, J. H. E. Proll, Yu. Turkin and P. Xanthopoulos
An overview is given of physics differences between stellarators and tokamaks, including magneto-hydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations performed for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.
Zonal Flow Dynamics and Control of Turbulent Transport in Stellarators
P. Xanthopoulos, A. Mischchenko, P. Helander, H. Sugama, and T.-H. Watanabe
The relation between magnetic geometry and the level of ion-temperature-gradient (ITG) driven turbulence in stellarators is explored through gyrokinetic theory and direct linear and nonlinear simulations. It is found that the ITG radial heat flux is sensitive to details of the magnetic configuration that can be understood in terms of the linear behavior of zonal flows. The results throw light on the question of how the optimization of neoclassical confinement is related to the reduction of turbulence.
Simulating gyrokinetic microinstabilities in stellarator geometry with GS2
J. A. Baumgaertel, E. A. Belli, W. Dorland, W. Guttenfelder, G. W. Hammett, D. R. Mikkelsen, G. Rewoldt, W. M. Tang and P. Xanthopoulos
The nonlinear gyrokinetic code GS2 has been extended to treat non-axisymmetric stellarator geometry. Electromagnetic perturbations and multiple trapped particle regions are allowed. Here, linear, collisionless, electrostatic simulations of the quasi-axisymmetric, three-field period national compact stellarator experiment (NCSX) design QAS3-C82 have been successfully benchmarked against the eigenvalue code FULL. Quantitatively, the linear stability calculations of GS2 and FULL agree to within~10%.
Oscillations of zonal flows in stellarators
P. Helander, A. Mishchenko, R. Kleiber and P. Xanthopoulos
The linear response of a collisionless stellarator plasma to an applied radial electric field is calculated, both analytically and numerically. Unlike in a tokamak, the electric field and associated zonal flow develop oscillations before settling down to a stationary state, the so-called Rosenbluth–Hinton flow residual. These oscillations are caused by locally trapped particles with radially drifting bounce orbits. The particles also cause a kind of Landau damping of the oscillations that depends on the magnetic configuration. The relative importance of geodesic acoustic modes and zonal-flow oscillations therefore varies among different stellarators.
Reducing turbulent transport in toroidal configurations via shaping
Recent progress in reducing turbulent transport in stellarators and tokamaks by 3D shaping using a stellarator optimization code in conjunction with a gyrokinetic code is presented. The original applications of the method focused on ion temperature gradient transport in a quasi-axisymmetric stellarator design. Here, an examination both of other turbulence channels and other starting configurations is initiated. It is found that the designs evolved for transport from ion temperature gradient turbulence also display reduced transport from other transport channels whose modes are also stabilized by improved curvature, such as electron temperature gradient and ballooning modes. The optimizer is also applied to evolving from a tokamak, finding appreciable turbulence reduction for these devices as well. From these studies, improved understanding is obtained of why the deformations found by the optimizer are beneficial, and these deformations are related to earlier theoretical work in both stellarators and tokamaks.
Up to now, the term “transport-optimized” stellarators has meant optimized to minimize neoclassical transport, while the task of also mitigating turbulent transport, usually the dominant transport channel in such designs, has not been addressed, due to the complexity of plasma turbulence in stellarators. Here, we demonstrate that stellarators can also be designed to mitigate their turbulent transport, by making use of two powerful numerical tools not available until recently, namely, gyrokinetic codes valid for 3D nonlinear simulations and stellarator optimization codes. Two initial proof-of-principle configurations are obtained, reducing the level of ion temperature gradient turbulent transport from the National Compact Stellarator Experiment baseline design by a factor of 2–2.5.
Signature of a universal statistical description for drift-wave plasma turbulence
This letter provides a theoretical interpretation of numerically generated probability density functionsPDFsof intermittent plasma transport events. Specifically, nonlinear gyrokinetic simulations of ion-temperature-gradient turbulence produce the time series of heat flux that manifestly exhibit non-Gaussian PDFs with enhanced tails. It is demonstrated that, after the remova lof autocorrelations, the numerical PDFs can be matched with predictions from a fluid theoretical setup based on the instanton method. This result points to a universality in the modeling of intermittent stochastic process offering a predictive capability.
Using the nonlinear gyrokinetic code package GENE/GIST [F. Jenko, W. Dorland, M.Kotschenreuther, and B. N. Rogers, Phys. Plasmas7, 19042000; P. Xanthopoulos, W. A. Cooper, F. Jenko, Yu. Turkin, A. Runov, and J. Geiger, ibid.16, 0823032009], we study the turbulent transport in a broad family of stellarator designs, to understand the geometry dependence of the microturbulence. By using a set of flux tubes on a given flux surface, we construct a picture of the two-dimensional structure of the microturbulence over that surface and relate this to relevant geometric quantities, such as the curvature, local shear, and effective potential in the Schrödinger-like equation governing linear drift modes.
A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations
P. Xanthopoulos, W. A. Cooper, F. Jenko, Yu. Turkin, A. Runov and J. Geiger
Physics of Plasmas 16, 082303 (2009)
The GENE/GIST code package is developed for the investigation of plasma microturbulence, suitable for both stellarator and tokamak configurations. The geometry module is able to process typical equilibrium files and create the interface for the gyrokinetic solver. The analytical description of the method for constructing the geometric elements is documented, together with several numerical evaluation tests. As a concrete application of this product, a cross-machine comparison of the anomalous ion heat diffusivity is presented.
Gyrokinetic turbulence under near-separatrix or nonaxisymmetric conditions
F. Jenko, D. Told, P. Xanthopoulos, F. Merz and L. D. Horton
Physics of Plasmas 16, 055901 (2009)
Linear and nonlinear gyrokinetic simulations with the GENE code [F. Jenko et al., Phys. Plasmas7,19042000] for tokamak edge plasmas as well as for stellarator core plasmas are presented, shedding light on the behavior of plasma microturbulence under near-separatrix or nonaxisymmetric conditions. To this aim, the required geometric coefficients are inferred directly from the magnetohydrodynamic equilibria of three different devices via the newly developed GIST code. It is found that the residual electron heat transport level in the H-mode edge can be explained in terms of high-wave-number fluctuations driven by electron temperature gradient modes. Moreover, the study of adiabatic ion temperature gradient turbulence in optimized stellarators points to the possibility of a systematic geometric optimization with respect to anomalous transport in nonaxisymmetric devices.
Verification and application of numerically generated magnetic coordinate systems in gyrokinetics
P. Xanthopoulos, D. Mikkelsen, F. Jenko, W. Dorland and O. Kalentev
In the context of linear gyrokinetic simulations, an analysis of the application of field-aligned coordinate systems generated numerically from magnetohydrodynamic equilibria is presented. This family of systems allows some flexibility in the choice of the coordinates, and gyrokinetic solvers often differ in this respect. Certain transformations are therefore required in order to compare physics results. Accordingly, benchmarks of a linear microinstability are carried out between two similar gyrokinetic codes. Effort is also put on the verification of the special properties of the generated systems through certain diagnostics.
Gyrokinetic microinstabilities in ASDEX Upgrade edge plasmas
D. Told, F. Jenko, P. Xanthopoulos, L. D. Horton, E. Wolfrum and the ASDEX Upgrade Team
Results of linear gyrokinetic simulations of ASDEX Upgrade [O. Gruber et al., Nucl. Fusion 39,1321 (1999)] edge plasmas, with experimentally determined geometry and input parameters, are presented. It is found that in the near-edge region, microtearing modes can exist under conditions found in conventional tokamaks. As one enters the steep-gradient region, the growth rate spectrum is dominated—down to very low wavenumbers—by electron temperature gradient modes. The latter tend to peak near the X-points and possess properties which may explain the ratios of the density and temperature gradient scale lengths that have been observed in various experiments over the last decade.
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves
An initial- and Dirichlet boundary value- problem for a Klein–Gordon–Schrödinger-type system of equations is considered, which describes the nonlinear interaction between high frequency electron waves and low frequency ion plasma waves in a homogeneous magnetic field. To approximate the solution to the problem a linearly implicit finite difference method is proposed, the convergence of which is ensured by deriving a second order error estimate in a discrete energy norm that is stronger than the discrete maximum norm. The numerical implementation of the method gives a computational confirmation of its order of convergence and recovers known theoretical results for the behavior of the solution, while revealing additional nonlinear features.
Nonlinear Gyrokinetic Simulations of Ion-Temperature-Gradient Turbulence for the Optimized Wendelstein 7-X Stellarator
P. Xanthopoulos, F. Merz, T. Görler, and F. Jenko
Physical Review Letters 99, 035002 (2007)
Ion-temperature-gradient turbulence constitutes a possibly dominant transport mechanism for optimized stellarators, in view of the effective suppression of neoclassical losses characterizing these devices. Nonlinear gyrokinetic simulation results for the Wendelstein 7-X stellarator [G. Grieger et al., in Proceedings of the IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, 1990 (IAEA, Vienna, 1991) Vol. 3, p. 525]—assuming an adiabatic electron response—are presented. Several fundamental features are discussed, including the role of zonal flows for turbulence saturation, the resulting flux-gradient relationship, and the coexistence of ion-temperature-gradient modes with trapped ion modes in the saturated state.
Gyrokinetic analysis of linear microinstabilities for the stellarator Wendelstein 7-X
A linear collisionless gyrokinetic investigation of ion temperature gradient (ITG) modes—considering both adiabatic and full electron dynamics—and trapped electron modes (TEMs) is presented for the stellarator Wendelstein 7-X W7-X [G. Grieger et al.,Plasma Physics and Controlled Nuclear Fusion Research 1990, International Atomic Energy Agency, Vienna, 1991, Vol. 3, p. 525]. The study of ITG modes reveals that in W7-X, microinstabilities of distinct character coexist. The effect of changes in the density gradient and temperature ratio is discussed. Substantial differences with respect to the axisymmetric geometry appear in W7-X, concerning the relative separation of regions with a large fraction of helically trapped particles and those of pronounced bad curvature. For both ITG modes and TEMs, the dependence of their linear growth rates on the background gradients is studied along with their parallel mode structure.
Clebsch-type coordinates for nonlinear gyrokinetics in generic toroidal configurations
The nonlinear gyrokinetic equations are frequently used as a basis for simulations of small-scale turbulence in magnetized toroidal plasmas. In this context, field-aligned coordinates are usually employed in order to minimize the number of necessary grid points. The present work proposes a system of Clebsch-type coordinates which does not depend on the existence of flux surfaces. The construction and use of these coordinates is explained, and the corresponding formulation of the nonlinear gyrokinetic equations is accomplished. This setup paves the way toward the investigation of non axisymmetric toroidal geometries, also in the region of magnetic islands as well as inside the ergodic layer where flux surfaces cease to exist. For testing purposes, in the axisymmetric, large aspect ratio case, the well-known expressions are recovered for closed flux surfaces. Moreover, geometric data for a specific stellarator configuration are computed and discussed.
Application of the 3D Finite Difference Scheme to the TEXTOR- DED Geometry
R. Zagorski, M. Jakubowski, N. McTaggart, R. Schneider, W. Stepniewski and P. Xanthopoulos
The effect of magnetic-ﬁeld-line ergodisation that eliminates poloidal magnetic surfaces imposes the need for plasma transport models being able to describe this properly. A prerequisite for a proper transport model is the concept of local magnetic coordinates allowing a correct discretization of the transport equations with minimized numerical errors. For the simulation of plasma transport in complex 3D edge geometries, a numerical method based on the ﬁnite difference concept has been developed, using a custom-tailored unstructured grid in local magnetic coordinates . This grid is generated by ﬁeld-line tracing to guarantee complete decoupling between the large parallel transport along B and that perpendicular to B. Therefore, the radial and parallel ﬂuxes can be effectively separated and the parallel and radial dynamics can be treated independently in the numerical method. In the FINITE code, the ﬁnite volume ansatz had been employed originally in order to resolve the radial transport on each Poincare surface.