Flux Surface Average Research Articles

Poloidal structure of the edge parallel flow in H-mode, L-mode and I-mode confinement regimes

Impurity profiles have been measured with the edge high field side (HFS) and low field side (LFS) charge exchange recombination spectroscopy suite at ASDEX Upgrade enabling the study of the poloidal structure of the edge parallel flow in H-mode, L-mode and I-mode. In H-mode, asymmetries in the impurity density, toroidal and poloidal rotations are found. In I-mode, only toroidal rotation asymmetries have been measured while in L-mode no asymmetries have been observed. The measured parallel flow can be divided in two components, the Pfirsch–Schlüter (PS) flow and the symmetric flow. Two different methods have been followed to determine both contributions to the parallel flow. The first method is based on the calculation of the PS flow at the HFS and LFS from the radial electric field. The second method directly provides the symmetric flow from the flux surface average (FSA) of the parallel flow. In H-mode, the methods provide different results, while in L-mode and I-mode they agree. The differences observed in H-mode between the two methods could be explained by the existence of asymmetries in the impurity density, by the non-negligible particle sources and radial losses, or by the approximations made in the calculation of the FSA of the parallel flow from measurements in two poloidal positions (midplane HFS and LFS) only.

Numerical evaluation of line, surface and toroidal integrals on level sets of toroidally symmetric functions

We investigate strategies to numerically integrate closed lines and surfaces that are implicitly defined by level sets (iso-contours) of continuously differentiable toroidally symmetric functions. The “grid-transform” approach transforms quantities given in non-surface-aligned coordinates onto a numerically constructed surface-aligned grid. Here, line and surface integrals, as well as so-called flux-surface averages, can be easily evaluated using high order integration formulas. We compare this method to ones that base on numerical representations of the delta-function.For the grid-transform method we observe high order convergence of line, surface and volume integration. Quantitatively, the errors for line and area integration are several orders of magnitude smaller than previously reported errors for delta-function methods. Furthermore, a delta-function method based on a Gaussian representation shows qualitatively wrong results of surface integrals near O- and X-points. Contrarily, the grid transform method suffers no deterioration near O-points. However, close to X-points we observe reduced first order convergence in volume integral and derivative tests due to the diverging volume element.Finally, we derive a toroidal integration based on toroidal summation and a smoothing kernel that assumes field-alignment of structures between the toroidal planes. The smoothing kernel can be interpreted as a partial flux-surface average. The resulting smoothed toroidal average eliminates unphysical poloidal oscillations that are otherwise present in the simple toroidal average.Our methods can be applied to toroidal and flux-surface averages in simulations of three-dimensional plasma dynamics on non-aligned grids. Further applications include closed line and surface integrals in level set methods. Efficient implementations can be found in the freely available Feltor library.

Stability of electron plasmas in stellarators

It has long been known that the equilibrium of an electron plasma in a stellarator possesses unique properties when compared with other geometries. Previous analyses, both numerical and analytical, as well as experimental results, have indicated that these equilibria are minimum-energy states. Here, it is definitively shown that the equilibrium of an electron plasma on magnetic surfaces with finite rotational transform minimises a constrained physical energy, which has a thermal and an electrostatic contribution. As such, these equilibria are established to be macroscopically stable to all perturbations that do not change the flux-surface average of the density and do not decrease the entropy of the plasma, under the definition of ‘formal stability’ established by Holmet al.(Phys. Rep., vol. 123, no. 1, 1985, 1–116).

Gyrokinetic simulations of neoclassical electron transport and bootstrap current generation in tokamak plasmas in the TRIMEG code

For magnetic confinement fusion in tokamak plasmas, some of the limitations to the particle and energy confinement times are caused by turbulence and collisions between particles in toroidal geometry, which determine the “anomalous” and the neoclassical transport, respectively. Neoclassical effects are also responsible for the intrinsically generated bootstrap current, and only the self-consistent modeling of neoclassical and turbulent processes can ultimately give accurate predictive results. In this work, we focus on the implementation of neoclassical physics in the gyrokinetic code TRIMEG, which is a TRIangular MEsh-based Gyrokinetic code that can handle both the closed and open field line geometries of a divertor tokamak. We report on the implementation of a simplified Lorentz collision operator in TRIMEG. For comparison with neoclassical theory, the calculation of flux surface averages is necessary. Since the code uses an unstructured mesh, a procedure for calculating the flux surface averages of particle and energy fluxes and the bootstrap current is derived without relying on the poloidal coordinate, which is useful also for other simulations in unstructured meshes. With the newly implemented collision operator, we study electron transport and bootstrap current generation in a plasma with a finite density gradient but uniform temperature for various simplified and realistic geometries. Compared with neoclassical theory, good agreement is obtained for the large aspect ratio case regarding the particle and energy fluxes as well as the bootstrap current. However, some discrepancies are observed at moderate aspect ratio and for a case with the realistic geometry of the ASDEX Upgrade tokamak. These deviations can be explained by different treatments and approximations in theory and simulation. In this paper, we demonstrate the capability to calculate the electron transport and bootstrap current generation in TRIMEG, which will allow for the self-consistent inclusion of neoclassical effects in gyrokinetic simulations in the future.

Effects of magnetic island on profile formation in flux-driven ITG turbulence

Full-f gyrokinetic simulations of ion temperature gradient (ITG) turbulence in the presence of a magnetic island are performed. A newly developed method for evaluating the flux-surface average is implemented to treat adiabatic electrons inside the magnetic island precisely. A neoclassical simulation below the threshold for linear ITG instability shows that the density profile does not relax at the O-point, although the ion temperature profile is flattened there. This results from the force balance in the direction of the magnetic field between the pressure gradient related to ion parallel motion and the mean radial electric field. A flux-driven ITG turbulence simulation shows a quasi-periodic transport reduction due to interaction between the background temperature profile and the vortex mode, which is a nonlinearly generated mesoscale structure with the same mode numbers as the magnetic island. These results indicate that not only the parallel streaming but also the equilibrium electric field and turbulence contribute significantly to profile formation around a magnetic island.

A Galerkin method for the weak formulation of current diffusion and force balance in tokamak plasmas

This paper presents a new numerical approach to the simulation of plasma equilibrium evolution in tokamaks. It is based on an unusual coupling formulation of current diffusion and axi-symmetric equilibrium equations that does not require the use of flux-surface averages and allows for fast Newton-type iterations schemes. A detailed derivation of the coupled system of non-linear equations and its discretization is provided. Numerical examples, based on an implementation in the free-boundary equilibrium code FEEQS, highlight feasibility and applicability.

Self-consistent perturbed equilibrium with neoclassical toroidal torque in tokamaks

Toroidal torque is one of the most important consequences of non-axisymmetric fields in tokamaks. The well-known neoclassical toroidal viscosity (NTV) is due to the second-order toroidal force from anisotropic pressure tensor in the presence of these asymmetries. This work shows that the first-order toroidal force originating from the same anisotropic pressure tensor, despite having no flux surface average, can significantly modify the local perturbed force balance and thus must be included in perturbed equilibrium self-consistent with NTV. The force operator with an anisotropic pressure tensor is not self-adjoint when the NTV torque is finite and thus is solved directly for each component. This approach yields a modified, non-self-adjoint Euler-Lagrange equation that can be solved using a variety of common drift-kinetic models in generalized tokamak geometry. The resulting energy and torque integral provides a unique way to construct a torque response matrix, which contains all the information of self-consistent NTV torque profiles obtainable by applying non-axisymmetric fields to the plasma. This torque response matrix can then be used to systematically optimize non-axisymmetric field distributions for desired NTV profiles.

Intrinsic plasma rotation and Reynolds stress at the plasma edge in the HSX stellarator

Using multi-tipped Langmuir probes in the edge of the HSX stellarator, the radial electric field and parallel flows are found to deviate from the values calculated by the neoclassical transport code PENTA for the optimized quasi-helically symmetric (QHS) configuration. To understand whether Reynolds stress might explain the discrepancy, fluctuating floating potential measurements are made at two locations in the torus corresponding to the low field and high field sides of the device. The measurements at the two locations show clear evidence of a gradient in the Reynolds stress. However, the resulting flow due to the gradient in the stress is found to be large and in opposite directions for the two locations. This makes an estimation of the flux surface average using a small number of measurement locations impractical from an experimental perspective. These results neither confirm nor rule out whether Reynolds stress plays an important role for the QHS configuration. Measurements made in configurations with the quasi-symmetry degraded show even larger flows and greater deviations from the neoclassically calculated velocity profiles than the QHS configuration while the fluctuation magnitudes are reduced. Therefore, for these configurations in particular, the Reynolds stress is most likely not responsible for the additional momentum.

Tokamak equilibrium reconstruction code LIUQE and its real time implementation

Equilibrium reconstruction consists in identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. The LIUQE code adopts a computationally efficient method to solve this problem, based on an iterative solution of the Poisson equation coupled with a linear parametrisation of the plasma current density. This algorithm is unstable against vertical gross motion of the plasma column for elongated shapes and its application to highly shaped plasmas on TCV requires a particular treatment of this instability. TCV's continuous vacuum vessel has a low resistance designed to enhance passive stabilisation of the vertical position. The eddy currents in the vacuum vessel have a sizeable influence on the equilibrium reconstruction and must be taken into account. A real time version of LIUQE has been implemented on TCV's distributed digital control system with a cycle time shorter than 200μs for a full spatial grid of 28 by 65, using all 133 experimental measurements and including the flux surface average of quantities necessary for the real time solution of 1.5 D transport equations. This performance was achieved through a thoughtful choice of numerical methods and code optimisation techniques at every step of the algorithm, and was coded in Matlab and Simulink for the off-line and real time version respectively.

Kinetic effects on a tokamak pedestal ion flow, ion heat transport and bootstrap current

We consider the effects of a finite radial electric field on ion orbits in a subsonic pedestal. Using a procedure that makes a clear distinction between a transit average and a flux surface average we are able to solve the kinetic equation to retain the modifications due to finite drift orbit departures from flux surfaces. Our approach properly determines the velocity space localized, as well as the nonlocal, portion of the ion distribution function in the banana and plateau regimes in the small aspect ratio limit. The rapid variation of the poloidal ion flow coefficient and the electrostatic potential in the total energy modify previous banana regime evaluations of the ion flow, the bootstrap current, and the radial ion heat flux in a subsonic pedestal. In the plateau regime, the rapid variation of the poloidal flow coefficient alters earlier results for the ion flow and bootstrap current, while leaving the ion heat flux unchanged since the rapid poloidal variation of the total energy was properly retained.

CENTORI: A global toroidal electromagnetic two-fluid plasma turbulence code

A new global two-fluid electromagnetic turbulence code, CENTORI, has been developed for the purpose of studying magnetically-confined fusion plasmas on energy confinement timescales. This code is used to evolve the combined system of electron and ion fluid equations and Maxwell equations in toroidal configurations with axisymmetric equilibria. Uniquely, the equilibrium is co-evolved with the turbulence, and is thus modified by it. CENTORI is applicable to tokamaks of arbitrary aspect ratio and high plasma beta. A predictor–corrector, semi-implicit finite difference scheme is used to compute the time evolution of fluid quantities and fields. Vector operations and the evaluation of flux surface averages are speeded up by choosing the Jacobian of the transformation from laboratory to plasma coordinates to be a function of the equilibrium poloidal magnetic flux. A subroutine, GRASS, is used to co-evolve the plasma equilibrium by computing the steady-state solutions of a diffusion equation with a pseudo-time derivative. The code is written in Fortran 95 and is efficiently parallelised using Message Passing Interface (MPI). Illustrative examples of output from simulations of a tearing mode in a large aspect ratio tokamak plasma and of turbulence in an elongated conventional aspect ratio tokamak plasma are provided.

On radial geodesic forcing of zonal modes

The elementary local and global influence of geodesic field line curvature on radial dispersion of zonal modes in magnetised plasmas is analysed with a primitive drift wave turbulence model. A net radial geodesic forcing of zonal flows and geodesic acoustic modes can not be expected in any closed toroidal magnetic confinement configuration, since the flux surface average of geodesic curvature identically vanishes. Radial motion of poloidally elongated zonal jets may occur in the presence of geodesic acoustic mode activity. Phenomenologically a radial propagation of zonal modes shows some characteristics of a classical analogon to second sound in quantum condensates.

A unified treatment of kinetic effects in a tokamak pedestal

We consider the effects of a finite pedestal radial electric field on ion orbits using a unified approach. We then employ these modified orbit results to retain finite E × B drift departures from flux surfaces in an improved drift-kinetic equation. The procedure allows us to make a clear distinction between transit averages and flux surface averages when solving this kinetic equation. The technique outlined here is intended to clarify and unify recent evaluations of the banana regime decrease and plateau regime alterations in the ion heat diffusivity; the reduction and possible reversal of the poloidal flow in the banana regime, and its augmentation in the plateau regime; the increase in the bootstrap current; and the enhancement of the residual zonal flow regulation of turbulence.

Equilibrium reconstruction for single helical axis reversed field pinch plasmas

Single helical axis configurations are emerging as the natural state for high-current reversed field pinch plasmas. These states feature the presence of transport barriers in the core plasma. Here we present a method for computing the equilibrium magnetic surfaces for these states in the force-free approximation, which has been implemented in the SHEq code. The method is based on the superposition of a zeroth-order axisymmetric equilibrium and of a first-order helical perturbation computed according to Newcomb's equation supplemented with edge magnetic field measurements. The mapping of the measured electron temperature profiles, soft x-ray emission and interferometric density measurements on the computed magnetic surfaces demonstrates the quality of the equilibrium reconstruction. The procedure for computing flux surface averages is illustrated, and applied to the evaluation of the thermal conductivity profile. The consistency of the evaluated equilibria with Ohm's law is also discussed.

Toroidal flow and radial particle flux in tokamak plasmas

Many effects influence toroidal flow evolution in tokamak plasmas. Momentum sources and radial plasma transport due to collisional processes and microturbulence-induced anomalous transport are usually considered. In addition, toroidal flow can be affected by nonaxisymmetric magnetic fields; resonant components cause localized electromagnetic toroidal torques near rational surfaces in flowing plasmas and nonresonant components induce “global” toroidal flow damping torque throughout the plasma. Also, poloidal magnetic field transients on the magnetic field diffusion time scale can influence plasma transport. Many of these processes can also produce momentum pinch and intrinsic flow effects. This paper presents a comprehensive and self-consistent description of all these effects within a fluid moment context. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfvén waves (Grad–Shafranov equilibrium and ion radial force balance), sound waves (pressure constant along a field line and incompressible flows within a flux surface), and ion collisions (damping of poloidal flow). Finally, plasma transport across magnetic flux surfaces is induced by the many second order (in the small gyroradius expansion) toroidal torque effects indicated above. Nonambipolar components of the induced particle transport fluxes produce radial plasma currents. Setting the flux surface average of the net radial current induced by all these effects to zero yields the transport-time-scale equation for evolution of the plasma toroidal flow. It includes a combination of global toroidal flow damping and resonant torques induced by nonaxisymmetric magnetic field components, poloidal magnetic field transients, and momentum source effects, as well as the usual collision- and microturbulence-induced transport. On the transport time scale, the plasma toroidal rotation determines the radial electric field for net ambipolar particle transport. The ultimate radial particle transport is composed of intrinsically ambipolar fluxes plus nonambipolar fluxes evaluated at this toroidal-rotation-determined radial electric field.

Intrinsic rotation driven by the electrostatic turbulence in up-down asymmetric toroidal plasmas

The transport of parallel momentum by small scale fluctuations is intrinsically linked to symmetry breaking in the direction of the magnetic field. In tokamaks, an up-down asymmetry in the equilibrium proves to be an efficient parallel symmetry breaking mechanism leading to the generation of a net radial flux of parallel momentum by the electrostatic turbulence [Y. Camenen et al., Phys. Rev. Lett. 102, 125001 (2009)]. This flux is neither proportional to the toroidal rotation nor to its gradient and arises from an incomplete cancellation of the local contributions to the parallel momentum flux under the flux surface average. The flux of parallel momentum then depends on the asymmetry of the curvature drift and on the extension of the fluctuations around the low field side midplane. In this paper, the mechanisms underlying the generation of the flux of parallel momentum are highlighted and the main dependences on plasma parameters investigated using linear gyrokinetic simulations.

Turbulent anti-resistivity and the zonal magnetic field dynamo in drift-ballooning turbulence

The effect of turbulent fluctuations on the zonal (flux surface averaged) magnetic field in drift-ballooning turbulence is shown to be equivalent to a turbulent anti-resistivity. The flux surface average of the convective nonlinearity in Ohm’s law is proportional to the flux surface average of the current. The coefficient of proportionality can be defined as a turbulent resistivity ηturb. The correlation of the flux surface average of the convective nonlinearity with the negative of the flux surface average of the current is nearly 100%. Because the convective nonlinearity is correlated with the negative of the current, and not the current, the turbulent resistivity is negative. The magnitude of ηturb is virtually identical to the magnitude of the collisional resistivity η, but opposite in sign, so that the total resistivity ηtotal=η+ηturb is nearly zero. The effect of the fluctuations is to balance the effect of collisional resistive diffusion. As a result, while the energy in the zonal flow increases to a large value as the fluctuations grow and saturate, the energy in the zonal magnetic field remains very small.

Adiabatic plasma equilibrium and application to a reconnection problem

The evolution of many plasma systems is adiabatic, i.e., plasma entropy is conserved in each magnetic flux tube. An apparently surprising result found recently from simulations of a forced magnetic reconnection problem (the “Newton Challenge” [J. Birn et al., Phys. Plasmas 13, 092117 (2006)]) is that even in the presence of a dissipative process such as reconnection, the entropy within a flux tube can still be approximately conserved, due to the strong localization of the dissipation. To address plasma equilibrium with such adiabatic constraints, a novel code has been developed that computes equilibria with entropy profile as input, using the alternating dimension method [H. Grad et al., Proc. Natl. Acad. Sci. USA 72, 3789 (1975)]. The code alternates between solving the two–dimensional (2D) Grad-Shafranov equation to obtain the field configuration (flux function A) from the pressure profile P(A) and a 1D ordinary differential equation that uses the entropy conservation to derive the pressure function P(A) from a flux surface average. As a particular application, the code is used to compute equilibria relevant to the Newton Challenge, with a grid reflecting the reconnected state topology (with an X and an O point). The equilibria found agree very well with late stages of magnetohydrodynamic simulations of the Newton Challenge. This agreement not only validates the new code, but also proves that the final state approached by the reconnection simulations is indeed an equilibrium quasiadiabatically connected with the initial state. The results also show a significant release of magnetic energy through reconnection. Finally, other potential applications of the new code, especially to adiabatic evolution of space plasmas, are discussed.

Neoclassical Ion Flow Due to Inductive Electric Field in Tokamaks

The influence of an inductive parallel electric field is usually neglected in the neoclassical transport theory for ions. In a recent paper, Catto et al. have claimed that the ion flow due to the inductive electric field is surprisingly large in the banana regime and it becomes comparable to the neoclassical ion flow driven by the ion temperature gradient. Their calculation relies on the use of the Kovrizhnikh and Hirshman–Sigmar model collision operators. The numerical factors for the expressions of ion flow velocities depend on the choice of model collision operators. Therefore, the following questions arise. How are their obtained numerical factors modified when we employ more accurate collision operator? Can we use the standard moment method of Hirshman for calculating the electric-field driven ion flow? The purpose of this short note is to answer these questions. A gyrophase averaged distribution function for species a is assumed to be slightly perturbed from the Maxwellian fa0 in an axisymmetric magnetic field B. Then, in the banana regime, the perturbed distribution function fa1 due to an inductive electric field E is identically zero for trapped particles and the perturbed distribution function for passing particles is obtained from hðB=vkÞ P b Cabð fa1; fb1Þi 1⁄4 ðea=TaÞhBEkifa0, where vk 1⁄4 v b and Ek 1⁄4 E b with b 1⁄4 B=B; ea and Ta are the charge and the temperature; and h i denotes a flux-surface average. The linearized Fokker-Planck collision operator for species a and b has the form Cabð fa1; fb1Þ Cabð fa1; fb0Þ þ Cabð fa0; fb1Þ, where Cabð fa1; fb0Þ and Cabð fa0; fb1Þ represent the Landau collision operators for the perturbed test and field particle distributions. We here use the following approximation for the collision operator Cabð fa1; fb1Þ 1⁄4 DabðvÞLð fa1Þ þ C abð f 1 a1; f 1 b1Þ þ D abðvÞf 1 a1 ; ð1Þ

Sheared-flow modes in toroidal geometry

Using a Fourier–Bessel representation for the fluctuating (turbulent) electrostatic potential, an equation governing the sheared-flow modes in toroidal geometry is derived from the gyrokinetic Poisson equation, where both the adiabatic and nonadiabatic responses of the electrons are taken into account. It is shown that the principal geometrical effect on sheared-flow modes of the electrostatic potential is due to the flux-surface average of 1/B, where B is the magnetic field strength.