MIT-IFS-Culham-Chalmers-Lodestar UCSD Joint Program in Plasma Edge Physics: 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.

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Tünde Fülöp has been appointed as an assistant professor in fusion plasma physics in the Department of Electromagnetics at Chalmers University of Technology as of January 2000. She began her new appointment by initiating a collaboration between Chalmers, the Culham Science Centre, Forschnungszentrum in Jülich, Germany, and the University of Gent, Belgium to investigate the role of impurities in the edge plasma. The collaboration started with a visit to Jülich by Per Helander (Culham) and Tünde, where Tünde presented an invited talk about their work on nonlinear neoclassical transport in an impure edge plasma.

Jim Hastie Visiting Scientist appointment at the PSFC has been extended for an additional year and a half so he will now be at MIT until the end of October, 2001. During his stay Jim has become involved in MHD studies of edge plasma and other topics of interest to Alcator C-Mod and the Levitated Dipole Experiment (LDX).

Oleg Pogutse is working at Culham, comparing his Drift-Alfvén Turbulence theory of the L-H transition with edge data from COMPASS-D. Alexei Dnevstrovskii is visiting Culham to work on implementing a model for ELMs based on peeling-ballooning modes in the ASTRA transport code.

Visitors to the PSFC in the last few months included: Dan McCarthy of Southeastern Louisiana University at the beginning of March to continue his collaboration with Peter Catto on modeling edge turbulence; Barrett Rogers (University of Maryland) on February 9th and 10th to present a PSFC seminar entitled "Turbulent Transport in the Tokamak Edge", Leonid Zakharov (PPPL) on March 10th to give a PSFC seminar with the title "Tokamaks with Lithium Walls", and Bill Dorland (University of Maryland) from April 11th to 12th to work with Martin Greenwald to make his codes available to Alcator C-Mod and present an informal talk on recent gyrokinetic turbulence simulations of the ion temperatur gradient mode.

Jim Myra (Lodestar) presented a talk on "Resistive X-Point Modes in Tokamak Boundary Plasmas" at the recent International Sherwood Fusion Theory Meeting at UCLA from March 27-29. Talks were also given by Howard Wilson (Culham) on the "Effect of the Polarisation Current on Tearing Mode Stability" and another by Jeff Freidberg (MIT) entitled "Can Differentially Rotating Resistive Walls Stabilize Killer MHD Modes?". The other edge relevant papers by joint program members where presented during the poster sessions.

Sergei Krasheninnikov (UCSD) has been asked to give an oral presentation at the 27th EPS Conference on Controlled Fusion and Plasma Physics in Budapest, Hungary from 12 - 16 June. The title of his talk will be "Molecular Effects in Plasma Recombination". He has also been invited to attend the Divertor Simulator Workshop at the FOM-instituut voor Plasmafysica "Rijnhuizen" on 18-19 of May, 2000 to discuss the scientific program for divertor simulator to be built there.

Colin Roach (Culham) attended the International Workshop on Confinement and the ITER Expert Group Meetings on confinement at Naka, Japan, 27-30 March to discuss the status of the International Profile Database.

Jim Myra attended the NSTX Research Forum at PPPL, January 30 - February 2 and presented a summary of recent edge and SOL stability work. Although NSTX configurations remains to be investigated in detail, it is likely that magnetic geometry considerations will be important for understanding boundary plasma turbulence in spherical tori.

Introduction: A MARFE is normally considered to be a non-linear stage of a radiation - condensation instability that is driven by low charge state (low Z) impurity radiation. Accordingly, the plasma temperature in the MARFE region should be of the order of 5-10 eV for such radiation to be efficient. However, recent experimental studies revealed new features of the phenomena. In particular, a higher plasma density, ne ~ 2-3¥1021 m-3, and lower temperature, Te ~ 0.7 eV, than previously inferred, were measured in the MARFE region in Alcator C-Mod tokamak [1]. Low Z impurity radiation becomes relatively inefficient for such low temperatures. Moreover, experiments in TEXTOR-94 [2, 3] showed, that in some cases (in particular, in the case of favorable conditions for plasma recycling at the wall) the main contribution to the radiation losses from a MARFE is from hydrogen. It was also found that there was strong plasma recombination in the MARFE region. All this implies, that at least in some cases, a MARFE can be a nonlinear stage of an ionization-recombination instability of hydrogen plasma [2, 3, 4].

Background: The physics of the instability can be understood by considering a model with hydrogen plasma in the presence of hydrogen neutrals occupying a half-space (the inside of a tokamak chamber, for example) with the magnetic field parallel to the wall of the chamber and an outward energy flux towards the wall. Plasma diffuses across the magnetic field, B, and recombines volumetrically and at the wall, the resulting neutrals diffuse inward from the wall and eventually become ionized. In such a model a temperature drop near the wall causes an increase in plasma density due to parallel plasma pressure balance along B. This, in turn, leads to an increase in the plasma flux to the wall and, accordingly, to an increase in the neutral flux from the wall. The extra neutrals participate in the recycling process and more energy is lost from the system due to neutral radiation, causing the temperature to drop even more.

Results and Discussion: We performed a qualitative analysis of the ionization-recombination instability to show that the ratio of the energy source to the plasma flux to the wall times some characteristic energy, p, plays a particularly important role in determining the equilibrium temperature as well as the characteristics of the ionization-recombination instability. For "large" p (which means high temperature near the wall) the system is stable, while for "small" p (low temperature) perturbations parallel to B with wavelengths in some intermediate range are unstable. Short wavelength perturbations are stabilized by plasma parallel heat conduction. Because of tokamak geometry, long-wavelength perturbations along the magnetic field correspond to short-wavelength perturbations perpendicular to the magnetic field direction, which are stabilized by neutral heat conduction. The theory allows us to estimate the range of unstable wavelengths and the instability growth rate, which is typically given by g ~ (Dpl/DN2), where Dpl is perpendicular plasma diffusion coefficient determined by anomalous transport and DN is the neutral penetration length. When p is smaller than some critical value pcrit, which depends on Dpl, the equilibrium does not exist and the system experiences a thermal collapse.

After the qualitative behavior of the system was understood, detailed numerical modeling was performed. We solved a closed set of one dimensional fluid plasma equations describing a simple hydrogen plasma immersed in a magnetic field and in the presence of volumetric ionization and recombination, as well as plasma recombination at the wall. In the absence of parallel motion the equations give the equilibrium profiles for the plasma and neutral density and temperature. Once the equilibrium solution was found, a set of time dependent linearized partial differential equations for the plasma and neutral densities, temperature and parallel plasma velocity perturbations was solved numerically. The numerical results were in good agreement with the qualitative estimates for the instability growth rate, range of unstable parallel wave numbers, and their dependence on the system parameters (in particular, the plasma diffusion coefficient). It was also found that volumetric plasma recombination, when present, plays an important role in determining the behavior of the instability. In particular, for the case when p was just above pcrit, the numerical computations show that the instability growth rate with volumetric recombination can be four times larger than without it.

References.

[1] B. Lipschultz, J. L. Terry, C. Boswell, A. Hubbard, B. LaBombard and D. A. Pappas, Phys. Rev. Lett. 81, 1007 (1998).

[2] U. Samm, M. Brix, F. Durodie, M. Lehnen, A. Pospieszczyk, J. Rapp, G. Sergienko, B. Schweer, M. Z. Tokar and B. Unterberg, J. Nucl. Mater. 269, 666 (1999).

[3] M. Z. Tokar, J. Rapp, D. Reiser, U. Samm, F. C. Schüller, G. Sergienko and P. C. de Vries, J. Nucl. Mater. 269, 958 (1999).

[4] S. I. Krasheninnikov, B. Lipschultz and J. L. Terry, 25th EPS Conference on Controlled Fusion and Plasma Physics, Abstracts of Invited and Contributed Papers, 1998, Prague, (European Physical Society, Petit-Lancy, 1998), p.441.

Effect of Strong Radial Variation of Finite Larmor Radius Modifications on Internal Ballooning Modes, by R. J. Hastie (MIT), Peter J. Catto (MIT and Lodestar) and J. J. Ramos (MIT).

Motivation: The ideal magnetohyrodynamic (MHD) theory of infinite toroidal mode number (n Æ ï) ballooning modes has often been adopted to interpret the stability of edge pressure gradients in tokamaks. It is, however, well recognized that physically relevant instabilities have finite mode numbers and that for these finite diamagnetic drift frequency effects must be taken into account, especially in the edge plasma region between the separatrix and the high density side of the pedestal. Tang, Dewar and Manickam [1] showed that the finite gyroradius stabilizing influence of a constant ion diamagnetic drift frequency, w*pi, can dramatically modify the internal ballooning stability condition. Treating w*pi as a constant is normally a reasonable assumption; however, it often fails in the pedestal region just inside the separatrix where the density and temperature vary strongly over very short distances (often a centimeter or less in high confinement mode plasmas). As a result, the formalism for internal ballooning modes in a tokamak [2, 3] is extended to retain the strong radial variation of the ion diamagnetic drift frequency characteristic of edge plasmas in the pedestal region.

Analysis: We adopt the WKB formalism of Refs. 1 and 3 by writing any perturbed quantity such as a displacement x as a slowly varying amplitude  times an exponential term as follows: , where the safety factor q is a flux function that we use as the radial variable and , with z and  the toroidal angle and infinite domain poloidal angle variables, respectively. We assume q to be a monotonically increasing function of the poloidal flux function, y, in the pedestal region. The unknown function  depends on q and is determined by both the radial quantization condition on the radial eigenmode structure and the solution to some lowest order ballooning mode equation containing the local eigenvalue . The wave vector  associated with S is given by

(1)

with , , and . Notice that the function  is related to the radial wavenumber of the eigenmode. In the absence of finite gyroradius effects the local eigenvalue l is just w2, with w the wave frequency and true, global eigenvalue. However, when finite gyroradius effects are retained, l becomes [4].

, (2)

where the global eigenvalue is the wave frequency w and Eq. (2) is to be viewed as an implicit equation for .

For a plasma that is ballooning unstable, solving the ballooning mode equation results in a radially localized region of instability (l < 0) and the radial extent of the mode structure associated with the unstable region and the global eigenvalue w must satisfy the WKB quantization condition [1, 3] which for the most unstable lowest radial eigenmode is

, (3)

with the loop integral indicating an integration over a complete circuit between the turning points () and back. The radial quantization condition is necessary because only the solutions of the ballooning mode equation whose radial wavenumber n satisfies Eqs. (2) and (3) are of interest as they are the only ones that will fit in the radial well defined by the local eigenvalue  < 0. When  is parabolic in q and , Eq. (3) recovers the 1/n correction of Ref. 2; however, for non-parabolic behavior the corrections are in general not of order n and need only be small for n >> 1. To extend the WKB analysis to radially varying w*pi we can no longer proceed as in Ref. 1 by completing the square on the right side of Eq. (2), seeking marginally stable solutions by replacing l with  in the ballooning mode equation, and then solving it and using Eq. (2) to determine the  to be used in Eq. (3) to determine the values of n at marginal stability. Instead we retain w*pi(q) and consider marginal stability by letting w Æ w + ig with w real, g > 0, and g Æ 0, then the imaginary part of Eq. (3) gives, upon Taylor expanding about g = 0, the real equation

, ()

where the w derivative can be brought inside the integral because the integration is between the turning points. For a constant w*pi Eq. (8) is satisfied when w is equal to w*pi/2 and so reduces to the result of Ref. 1, while for w*pi radially varying w will be some suitably averaged version of w*pi.

Results: Our procedure for finding finite gyroradius modifications due to a radially varying diamagnetic drift frequency has been made more explicit by considering the s, a form of the ballooning mode equation, where s = (r/q)dq/dr is the shear parameter, a = - (2q2R/B2)dp/dr is the pressure gradient drive term. In the absence of finite gyroradius corrections the WKB procedure can give a rather low critical n = n1 for marginality (w = 0) with all n above this critical value unstable. With a constant w*pi another higher critical n = n2 is introduced above which ballooning modes are finite gyroradius stabilized. The unstable band of n1 < n < n2 is then modified further by the radial variation of w*pi. For an incompressible MHD treatment to be strictly valid the marginal frequencies w1 and w2 that correspond to n1 and n2, respectively, must be small compared to the sound frequency. In the absence of finite gyroradius effects the large n WKB procedure has been extended to the compressible limit by Dewar and Glasser [5]. In this compressible generalization the second order ballooning mode differential equation is replaced by two coupled second order differential equations, but the WKB part of the procedure is unchanged, so our finite gyroradius procedure remains applicable.

Motivation: The Spitzer-Härm [1] electron heat transport coefficient begins to fail even at small values of the ratio g defined as the mean free path of a thermal particle lth(cm) Å 1012T2(eV)/N(cm-3) (with T and N the plasma temperature and density) over the parallel temperature scale length. The Spitzer-Härm electron heat transport coefficient, in particular, typically fails for g > 1/100. The breakdown of fluid treatments occurs at these large values of collisionality because the mean free path increases as the square of the energy. This rapid increase causes the heat conduction to be dominated by particles with energies on the order of seven times the thermal energy and energy weighted mean free paths roughly 50 times larger than that of the thermal particles. As a result, weakly collisional energetic particles have a strong influence on parallel plasma transport.

Technique: To investigate long mean free path modifications of the electron heat conductivity the approach of Krasheninnikov [2] is adopted by seeking self-similar solutions of the high speed expansion of the full electron collision operator. However, to simplify the collision operator further the perpendicular distribution is assumed Maxwellian and an integration over perpendicular speeds is employed to obtain a collision operator depending only on the parallel velocity [3]. A self-similar solution of the electron kinetic equation for this model collision operator retains modifications of the parallel electron distribution due to electrons having energies E such that gE2/T2 ~ 1 (in this case the mean free path of an electron of energy E is l = lthE2/T2). Non-expandable, exponentially small modifications to the heat conduction, proportional to exp(-1/g1/2), that cannot be retained in conventional short mean free path treatments (g << 1) are shown to be responsible for the increase of the parallel heat flux above its Spitzer value. In addition, for most temperature profiles the actual value of the heat flux tends to be less than the Spitzer-Härm value when g  1/100 because the short mean free path expansion begins making the distribution function negative in significant regions of velocity space, thereby improperly making the Spitzer-Härm value larger than the actual value. Which of these processes dominates depends on the details of the temperature profile.

Results: The qualitative features are illustrated by considering the artificial problem in which the mean free path is proportional to E/T, rather than (E/T)2. In this special case we are able to solve analytically the self-similar kinetic equation analytically as well as numerically to determine the parallel heat flux. Both the artificial and physical cases are characterized by a family of equilibrium temperature profiles which depend on the single parameter a > 2. As a Æ 2 the temperature varies more strongly and the distribution develops an extended tail on the side having / > 0 that enhances the heat flux substantially above the Spitzer-Härm value, where  and  are the parallel electric field and the parallel velocity. A high energy tail is formed on one side of the distribution function and a depleted tail on the other because the mean free path increases with the energy squared so that electrons moving toward the cooler (warmer) region come from a warmer (cooler) region and have a longer (smaller) mean free path. The electric field further enhances this nonlocal behavior effect and the an extended algebraic tail becomes unnormalizable at a = 2. Away from a Æ 2 the temperature variation is weaker and the actual parallel heat flux is reduced from the Spitzer-Härm value which overestimates the heat flux because the short mean free path procedure allows the distribution function to become negative on the side with / < 0 .

References