The members of the Joint Program are pleased to welcome the Department of Mechanical and Aerospace Engineering at the University of California, San Diego as the newest member of the Joint Program in Plasma Edge Physics. The UCSD membership formally recognizes the continuation of a collaboration that began when Prof. Sergei I. Krasheninnikov was at the Plasma Science and Fusion Center (PSFC) of the Massachusetts Institute of Technology. Plasma edge theorists at the MIT PSFC in Cambridge; the Institute for Fusion Studies (IFS) at the University of Texas at Austin; the EURATOM/UKAEA Fusion Association at the Culham Science Centre in England; the Department of Electromagnetics at Chalmers University of Technology in Göteborg, Sweden; Lodestar Research Corporation, headquartered in Boulder, Colorado; and the Department of Mechanical and Aerospace Engineering at the University of California, San Diego participate in the Joint Program to stimulate and perform innovative collaborative research on edge physics.

Joint Program News
Meeting News
Research Reports

Resistive X-point Modes in Tokamak Boundary Plasmas, by J. R. Myra (Lodestar), D. A. D'Ippolito (Lodestar), X. Q. Xu (LLNL) and R. H. Cohen (LLNL)

 

Effect of Edge Convection on the H-Mode, by D. A. D'Ippolito and J. R. Myra (Lodestar), to appear in Proceedings of the Maastricht EPS Meeting

 

Dynamics of Runaway Electrons in Tokamak Disruptions, byP. Helander (Culham), F. Andersson (Chalmers) and L.-G. Eriksson (Cadarache)

 

Joint Program News

Sergei Krasheninnikov moved from Cambridge to San Diego in August to become a professor in the Department of Mechanical and Aerospace Engineering at the University of California, San Diego. Sergei presented invited talks at the 1999 meeting of the Atomic, Molecular and Optical Physics Program in Boulder, CO (October 24th - 27th) hosted by the Office of Basic Energy Sciences of DoE and at this years Annual Meeting of the Swedish Fusion Research Unit in Göteborg on November 8th and 9th. While in Göteborg he served as the outside representative on Tünde Fülöp's (Chalmers) Ph. D. dissertation committee on November 5th. Tünde's dissertation research was partially performed during her visits to the PSFC and Culham. Xavier Bonnin left the PSFC at the end of August to work in Greifswald in the German stellarator program.

Jim Hastie (Culham) will be a Visiting Scientist at the MIT PSFC for six months starting in November. During his visit Jim is expected to be involved in MHD studies of edge plasma and other topics of interest to Alcator C-Mod. Per Helander (Culham) visitied MIT PSFC from October 5 - 25th to collaborate with Dieter Sigmar (MIT) and Peter Catto (MIT/Lodestar) on transport related issues of interest to MAST, Compass, and Alcator C-Mod. During his visit Per presented a PSFC seminar on "Nonlinear Neoclassical Theory for the Tokamak Edge"and had extensive discussions with the C-Mod staff on the behavior of impurities in an attempt to explain the latest C-Mod observations. Magnus Grinneback (Chalmers) began a two and half month visit to the MIT PSFC on October 12th. Magnus will be working with Peter Catto on long mean free path effects on electron behavior. Magnus is a student of Mietek Lisak and Dan Anderson at Chalmers.

Other visitors to the PSFC included Dan McCarthy of Southeastern Louisiana University at the end of Augustto continue his collaboration with Peter Catto and Sergei Krasheninnikov on modeling edge turbulence; Frank Waelbroeck (IFS) on October 15th and 16th to present a PSFC seminar entitled "Role of the Polarization Current in Neoclassical Tearing Modes", and Bill Dorland (University of Maryland) from September 15th to 17th to present a PSFC seminar on "Electron Temperature Gradient Driven Turbulence".

Jim Myra (Lodestar) attended the Plasma Edge Theory (PET-7) Workshop in Tajimi, Japan from October 4 - 6th and gave an invited review talk on "MHD and Fluid Instabilities at the Plasma Edge in the Presence of a Separatrix and X-Point." The PET-7 meeting papers will be published in a special issue of Contributions to Plasma Physics.

Dan D'Ippolito (Lodestar), Per Helander (Culham), Peter Catto and Sergei Krasheninnikov attended the Twenty-Sixth EPS Conference on Controlled Fusion and Plasma Physics in Maastricht, The Netherlands, from June 14th-18th 1999 where they presented their latest work in oral and contributed papers. The papers are published in the EPS proceedings which are available on line at http://epsppd.epfl.ch/.

Dan also attended the 2nd Europhysics Workshop on the Role of Electric Fields in Plasma Confinement and Exhaust in Maastricht from June 19th to 20th, and presented a longer paper on the Lodestar edge convection work. This paper will be published in the REFPCE proceedings in a CD-ROM supplement to the Czech. Journal of Physics.

Jack Connor, Per Helander and Chippy Thyagaraja (Culham) attended and presented papers at the IAEA TCM on First Principles Transport Theory at Kloster Seeon, Germany, 21-23 June. Jack's was entitled "Radially Non-local Effects on the Structure of Plasma Profiles at the Edge of a Tokamak", while Per's was "Nonlinear Neoclassical Transport for the Tokamak Edge".  Jack Connor, Colin Roach and Marion Turner(Culham) attended the International Confinement Meeting held at JET and Culham, 30 September - 1 October.
 

Motivation: The presence of a magnetic X-point and divertor is known experimentally to influence the tokamak boundary plasma and L-H transition. While the X-point and divertor influence many aspects of the boundary plasma equilibrium, one important effect, considered here, is the effect of the magnetic geometry on unstable modes and edge turbulence. In particular, along a field line the X-point introduces a region of strong magnetic shear and increased connection length.

Analysis: The nonlocal electromagnetic 3D BOUT code has enabled 3D turbulence simulations in realistic divertor geometry including both the edge and scrape-off-layer (SOL) regions [1, 2]. In combination with the linear eikonal BAL code [3] the linear and nonlinear behavior of modes described within the reduced Braginskii model have been studied.

Using experimentally measured density and temperature profiles for DIII-D as base case input [1, 2], we have identified several possible instability branches and drives: ideal MHD, resistive MHD, drift-Alfven, and SOL modes associated with sheath boundary conditions (BCs). The simulations indicate that a particular type of resistive ballooning mode which we refer to as the resistive X-point (RX) mode is the dominant instability for the L phase discharge.

The basic physics properties of the RX mode are best understood from the ballooning formalism. In lowest order, k*B = 0 = k_zB_z + k_qB_q yields kq = nB_z/RB_q, which is singular at the X-point where B_qÆ 0. Here our orthogonal coordinate system is (y, q, z) = (radial, poloidal, toroidal), and n is the toroidal mode number. In eikonal theory, the ey component of k may be obtained from e_z*óxk = 0 (since k =óS for some eikonal function S) and this yields and integral for ky which contains the singular kq in the integrand. The growth of ky is controlled by magnetic shear, as usual in the ballooning formalism. The magnetic shear effect of the X-point, however, is dramatic and quantitatively unlike the shear effect present for tokamaks without a separatrix. The rapid growth of k_y along a field line passing near the X-point may be viewed as arising from the strong squeezing deformation of flux tubes passing near the X-point [4]. The enhancements of both k_q and k_y give rise to enhancements of k_^² which can easily exceed an order of magnitude, for a single pass at distances of several ion Larmor radii from the X-point. Thus, the increased importance of effects proportional to k_^² such as ion finite Larmor radius effects (FLR), electron resistivity and inertia, near the X-point can be anticipated. One effect of an X-point is therefore to decouple field line regions on either side of it [4].

A typical unstable spectrum in our L-phase BAL runs is dominated by two distinct types of curvature-driven resistive ballooning mode instabilities. One branch peaks at high n and is identified as the classical resistive ballooning mode [5], the other peaks at low n and is the RX mode. The underlying physics of the RX mode and the classical resistive ballooning mode is similar - both modes decay away from the outboard midplane (with reduced MHD line-bending penalty) due to resistivity, made significant by the magnetic-shear-induced growth of k_y. The classical resistive ballooning mode, which is relatively localized to the outboard midplane, requires high n to make resistive effects µhk_^²significant. In contrast, the RX mode exists for much lower n because k_^²increases by many orders of magnitude near the X-point. Eigenfunctions for the RX mode are somewhat interchange-like on the bad-curvature-side of the torus, and decay to zero when they encounter the X-point region; the electrostatic potential normally peaks near the X-point region where the mode transitions from being electromagnetic to electrostatic.

Conclusion: X-point geometry allows a new possibility of fast growing resistive instabilities at moderate n. This new mode occurs because the effects of X-point geometry and resistivity are synergistic. These RX modes are more important for turbulent transport than their high-n classical resistive ballooning mode counterparts, as can be seen from the mixing length estimate D ~ g/k_^² and from the simulations. Nonlinear BOUT simulations with fixed profiles have shown turbulent suppression in the H phase relative to the L phase and calculated ion thermal diffusivities comparable to those in the experiment [1]. Recently a dynamic evolution of the L-H transition has been obtained, and will be reported elsewhere [6].

Background: This work is motivated by a number of experimental observations, including the following: (i) steady state convective flows have been measured in the edge and SOL of several tokamaks, caused by spatially localized disturbances such as rf antennas and gas puffing; (ii) JET data [1-4] shows that H-mode properties such as the ratio of tE/tp (energy over particle confinement time), the temperature pedestal height, and the ELM amplitude and repetition rate can be significantly different for ICRF H-modes under certain conditions than for NBI H-modes; (iii) there is an interesting parallel between the effects of ICRF and gas puffing on the H-mode in JET; and (iv) recent measurements on Alcator C-Mod [5] showing that convection may be responsible for a significant fraction of the energy transport across the separatrix. It has been shown that under certain circumstances (favorable to the formation of rf sheaths) ICRF can drive strong convection in the SOL and edge of a tokamak [2],and it has been postulated [1-4] that this convection is the mechanism responsible for some of the observations in (ii) and (iii).More generally, other sources of poloidal non-uniformity may drive convection, which may modify the H-mode properties in significant ways. These observations have motivated the development of a theory of edge convection and its effect on the temperature pedestal and transport barrier in the H-mode.

Results: For collisional plasmas described by the Braginkii equations, we have derived a set of nonlinear model equationsdescribing the interaction of E¥B convection with the tokamak edge plasma electric field and electron temperature T_e. The convection can be driven by a poloidal modulation of either the edge potential f (e.g. due to ICRF-driven sheath effects) or the edge T_e (e.g. due to gas puffing). The symmetry between the perturbations of f and T_ein the theory (due to the thermoelectric effect), may relate to the observed parallels between the effects of ICRF and gas puffing on H-mode confinement. The nonlinear interaction between the (zero-frequency) perturbations of f and T_e gives rise to additional terms in the vorticity and heat diffusion equations; in the limit of strong convection, the nonlinear terms dominate the equilibrium and impose constraints relating the equilibrium flux-surface-averaged radial electric field, E_x,and electron temperature, T_e, profiles which are not present in the usual H-mode. An analytic solution for typical boundary conditions shows that strong DC convection can produce significant cooling, reduction in the E¥B shear, and reversal of the sign of E_x in the edge plasma inside the separatrix.

Combined with the SOL model of rf-driven convection in Ref. 2, this work provides a mechanism to explain the experimental dependence of the ICRF H-mode on the JET antenna phasing [1,2]. It may also be relevant to the other experimental observations mentioned above. Convective cooling can reduce the temperature pedestal, increase the edge plasma resistivity, and thereby change the character of the MHD modes producing ELMs; the modified E¥B shear affects the edge turbulence and the global confinement, if the convection penetrates into the transport barrier region; also, the sign reversal in E_x should reduce the ion confinement in the convective layer. For example, the H-mode biasing experiments on Textor [6] showed a clear asymmetry between H-modes produced with positive and negative E_x: comparable t_E in the two cases, but the ratio of t_p/t_E was about three times lower for E_x > 0. A similar reduction in t_p was obtained in the "low particle confinement" H-modes on JET[7] and the "Enhanced D_a" H-modes on C-Mod [8]. In future work, we will extend this model to include non-rf-driven convection, density evolution and particle transport, and examine in more detail the relation of the theory to these experiments.

More detailed reports [9, 10] on this work can be downloaded from the Lodestar web site.

References

Introduction: Runaway electrons produced in tokamak disruptions can be a severe problem since their loss to the first wall may cause localised surface damage. The situation is particularly serious in large tokamaks, where avalanches of runaway electrons can be created by close Coulomb collisions with thermal electrons [1]. It is therefore important to understand the physics behind the generation and loss of runaway electrons. The present work is devoted to two topics in this area: damping of runaway current by synchrotron radiation, and avalanche mitigation by radial runaway diffusion.

Emission of synchrotron radiation: In JET, a large (about 1 MA) runaway current sometimes persists long after a disruption, showing a smooth decay on a time scale of one or two seconds. This decay cannot be explained by collisional drag alone, and it has been proposed that it could be caused by the emission of synchrotron radiation [2].

The velocity vector of a runaway electron is initially nearly parallel to the magnetic field, but needs only be scattered slightly to acquire Larmor rotation that leads to substantial synchrotron radiation. Since the radiation from a relativistic particle is emitted in a beam centred around the velocity vector, the reaction force is mainly in the parallel direction although it is the perpendicular motion that causes the radiation.

To describe the kinetics of this problem we solve a Fokker-Planck equation which includes collisional drag, pitch-angle scattering, radiation reaction, and a parallel electric field. The calculation is done for a beam of strongly relativistic electrons, and the electric field is induced by the current decay in a self-consistent way. The solution shows that synchrotron radiation emission indeed damps the runaway current more effectively than the collisional drag force, and the damping time scale appears consistent with the measurements.

Radial diffusion: In order to address the question whether sufficiently strong radial diffusion can interrupt a runaway avalanche, we derive a simple integral equation that accounts for the simultaneous creation (by close collisions), acceleration, and loss (by radial diffusion due to magnetic fluctuations) of runaways. This equation provides the means to calculate the avalanche growth rate in a situation where the runaways are subject to radial diffusion but become progressively better confined as they are accelerated by the electric field. In order to suppress runaway avalanches, their growth rate must by reduced to a level where there is not enough induced electric field to achieve significant avalanche growth. The amplitude of the magnetic fluctuations required for this depends sensitively on their mode structure.

We have constructed a three-dimensional Monte Carlo code that numerically solves the orbit-average of the kinetic equation for runaway electrons in toroidal geometry. The effects of collisional drag, pitch-angle scattering, parallel electric field, synchrotron radiation reaction, and radial diffusion are all included in the simulation. A source of new runaways is provided by the inclusion of the leading-order term in the quantum mechanical Møller scattering formula for close collisions between runaway electrons and thermal ones.

When there is no radial diffusion, the calculated growth rate of runaway avalanches agrees well with Ref 1. When enough radial diffusion is added, so that the runaway confinement time becomes comparable with the growth rate, the latter falls noticeably. In agreement with analytical results, if there is no synchrotron radiation the growth rate approaches zero asymptotically, but always stays positive, as the diffusion coefficient increases. However, when the radiation reaction force is included in the calculation, the growth rate vanishes for some finite (but large) diffusion coefficient. In either case, enough radial diffusion effectively prevents the runaway avalanche.

References

[1] M.N. Rosenbluth and S.V. Putvinski, Nucl. Fusion 37, 1355 (1997).

[2] R.D. Gill, Nucl. Fusion 33, 1613 (1993).