Thesis for the degree of Doctor of Philosophy. Turbulent and neoclassical transport in tokamak plasmas. István Pusztai

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1 Thesis for the degree of Doctor of Philosophy Turbulent and neoclassical transport in tokamak plasmas István Pusztai Nuclear Engineering Department of Applied Physics Chalmers University of Technology Göteborg, Sweden, 2011

2 Turbulent and neoclassical transport in tokamak plasmas István Pusztai c István Pusztai, 2011 ISBN Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 3262 ISSN X Department of Applied Physics Chalmers University of Technology SE Göteborg Sweden Telephone +46 (0) Printed in Sweden by Reproservice Chalmers Tekniska Högskola Göteborg, Sweden, 2011

3 Turbulent and neoclassical transport in tokamak plasmas István Pusztai Department of Applied Physics Chalmers University of Technology Abstract One of the greatest challenges of thermonuclear fusion is to understand, predict and to some extent control particle and energy transport in fusion plasmas. In the present thesis we consider theoretical and experimental aspects of collisional and turbulent transport in tokamak plasmas. First the collisionality dependence of quasilinear particle flux due to ion temperature gradient (ITG) and trapped electron modes is investigated. A semi-analytical gyrokinetic model of electrostatic microinstabilities is developed and used to study various parametric dependences of ITG stability thresholds and quasilinear particle and energy fluxes, focusing on the effect of collisions. Then corrections to the neoclassical plateau regime transport in transport barriers are calculated. It is found that the ion temperature gradient drive of the bootstrap current can be enhanced significantly, and the ion heat diffusivity and the poloidal flow of trace impurities are also modified in the presence of strong radial electric fields. Furthermore, we investigate the characteristics of microinstabilities in electron cyclotron heated and ohmic discharges in the T10 tokamak using linear gyrokinetic simulations, aiming to find insights into the effect of auxiliary heating on the transport, with special emphasis on impurity peaking. The effect of primary ion species of differing charge and mass on instabilities and transport is studied through linear and nonlinear gyrokinetic simulations. We perform transport analysis of three balanced neutral beam injection discharges from the DIII-D tokamak which have different main ion species (deuterium, hydrogen and helium). Finally the magnitude and characteristics of the error in alkali beam emission spectroscopy density profile measurements due to finite beam width are analyzed and a deconvolution based correction algorithm is introduced. Keywords: thermonuclear fusion, tokamaks, microinstabilities, turbulent transport, neoclassical transport, transport barriers, impurity transport, isotope effect, transport analysis iii

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5 Publications [A] T. Fülöp, I. Pusztai, and P. Helander Collisionality dependence of the quasilinear flux due to microinstabilities, Phys. Plasmas 15, (2008). [B] I.Pusztai,T.Fülöp, J. Candy, and R. J. Hastie, Collisional model of quasilinear transport driven by toroidal electrostatic ion temperature gradient modes, Phys. Plasmas 16, (2009). [C] I. Pusztai and P. J. Catto, Neoclassical plateau regime transport in a tokamak pedestal, Plasma Phys. Control. Fusion 52, (2010). [D] I. Pusztai, S. Moradi, T. Fülöp, and N. Timchenko, Characteristics of microinstabilities in electron cyclotron and ohmic heated discharges, Phys. Plasmas 18, (2011). [E] I. Pusztai, J. Candy, and P. Gohil, Isotope mass and charge effects in tokamak plasmas, accepted for publication in Phys. Plasmas. [F] I. Pusztai, G. Pokol, D. Dunai, D. Réfy, G. Pór, G. Anda, S. Zoletnik, and J. Schweinzer, Deconvolution-based correction of alkali beam emission spectroscopy density profile measurements, Rev. Sci. Instrum. 80, (2009). v

6 Other contributions (not included in the thesis) [G] P. J. Catto, G. Kagan, M. Landreman, and I. Pusztai, A unified treatment of kinetic effects in a tokamak pedestal, Plasma Phys. Control. Fusion 53, (2011). [H] T. Fülöp, S. Braun, and I. Pusztai, Impurity transport driven by ion temperature gradient turbulence in tokamak plasmas, Phys. Plasmas 17, (2010). [I] S. Moradi, T. Fülöp, A. Mollén, and I. Pusztai, A possible mechanism responsible for generating impurity outward flow under radio frequency heating, Plasma Phys. Control. Fusion 53, (2011). [J] I. Pusztai, J. Candy, P. Gohil, and E. A. Belli, Isotope mass and charge effects in tokamak plasmas, Proceedings of the 38th EPS Conference on Plasma Physics, P (2011). [K] T. Fülöp, S. Moradi, A. Mollén, and I. Pusztai, The role of poloidal asymmetries in impurity transport, Proceedings of the 38th EPS Conference on Plasma Physics, P (2011). [L] D. Guszejnov, G. I. Pokol, I. Pusztai, and D. Réfy, Applications of the RENATE beam emission spectroscopy simulator, Proceedings of the 38th EPS Conference on Plasma Physics, P (2011). [M] I. Pusztai and P. J. Catto, Neoclassical plateau regime transport in a tokamak pedestal, Proceedings of the 37th EPS Conference on Plasma Physics, ECA Vol.34A, P (2010). [N] T. Fülöp, S. Braun, and I. Pusztai, Impurity transport driven by electrostatic turbulence in tokamak plasmas, Proceedings of the 37th EPS Conference on Plasma Physics, ECA Vol.34A, P (2010). [O] T. Fülöp, S. Braun, and I. Pusztai, Impurity transport driven by electrostatic turbulence in tokamak plasmas, Proceedings of 23rd IAEA Fusion Energy Conference, pp. THC/P4-09 (2010). [P] S. Kálvin, G. Anda, D. Dunai, G. Petravich, S. Zoletnik, G. Pokol, B. Játékos, I. Pusztai, and D. Réfy, Reconstruction of plasma vi

7 edge density profile from Lithium beam data using statistical analysis, Proceedings of the 36th EPS Conference on Plasma Physics, ECA Vol.33E, P (2009). [Q] I. Pusztai, T. Fülöp, and P. Helander, On the quasilinear transport fluxes driven by microinstabilities in tokamaks, Proceedings of the 35th EPS Conference on Plasma Physics, ECA Vol.32D, P (2008). [R] T. Fülöp, I. Pusztai, and P. Helander, Quasilinear transport fluxes driven by electrostatic microinstabilities in tokamaks, Proceedings of 22nd IAEA Fusion Energy Conference, pp. TH/P8-28. (2008). [S] I. Pusztai, D. Dunai, G. Pokol, G. Pór, J. Schweinzer, and S. Zoletnik, Capabilities of alkali Beam Emission Spectroscopy for density profile and fluctuation measurements, Proceedings of the 34th EPS Conference on Plasma Physics, ECA Vol.31F, P (2007). vii

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9 Contents Abstract Publications Acknowledgments iii v xi 1 Introduction 1 2 Basic concepts in the transport of magnetized plasmas Magnetized fusion plasmas Magnetic geometry and particle motion Distribution function and kinetic equations Neoclassical transport Collisional transport across flux surfaces Drift kinetic equation Collisionality regimes Bootstrap current Neoclassical transport in H-mode pedestals Turbulent transport and microinstabilities Gyrokinetic description Ballooning formalism Particle and heat fluxes Microinstabilities Ion temperature gradient mode Trapped electron mode The role of collisions in turbulent transport Nonlinear simulations and transport analysis ix

10 5 Beam emission spectroscopy Turbulence measurements Electron density measurements Summary 61 References 65 Included papers A-F 71 x

11 Acknowledgments First of all I would like to thank my supervisor, Tünde Fülöp with her kindness, brilliance and exceptional intuitiveness, for her support, encouragement and valuable guidance in research and in other areas of life. I am very grateful to my MSc supervisor, Gergő Pokol, who started me on this path, for his support, guide and friendship. Special thanks go to my temporary or external supervisors: I would like to thank Jeff Candy with his explanatory talent, for the fruitful discussions. I am indebted to Peter Catto who always found the time for sharing his sharp insights to physical problems. And finally, the help and instructive guidance of Per Helander is greatly acknowledged. I am really thankful for the Fusion Theory group at General Atomics and the Plasma Science and Fusion Center at MIT for supporting my visits and for their hospitality. I am very grateful to all my colleagues at Nuclear Engineering for providing a welcoming stimulating atmosphere, especially to professor Imre Pázsit and to Albert, Geri and Sara from the Fusion Theory group. I will always remember the good old days at the group of Nonlinear Electrodynamics, where I am thankful for professors Mietek Lisak and Dan Anderson, and the graduate students Robert, Tobias and Joel for providing an inspiring atmosphere. I would also like to thank all my coauthors and collaborators for their help. But above all, I owe my warmest gratitude to my little family; my wife Renáta, for her patience, encouragement and love, and my daughter, Isgerdur. xi

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13 Chapter 1 Introduction The ever increasing energy demand of humanity, our finite non-renewable resources and the threatening extent of environmental pollution establish the need for a new, clean and large-scale energy source. One of the most promising candidates for this purpose is controlled thermonuclear fusion which has been an intensely explored area for more than half a century. Fusion utilizes the energy that is released as two light nuclei fuse together, and provides one of the main energy sources of the Universe. The reaction between the positively charged nuclei is obstructed by their Coulomb repulsion, so that bringing even the most feasible fusion process [D + T 4 He (3.5 MeV)+n(14.1 MeV)] to effect with reasonable efficiency (at achievable density) requires a temperature of 10 8 K, which implies that in laboratory conditions the fusion fuel has to be confined by some special means. Fortunately, the extremely high temperatures giving rise to the difficulties of confinement also provide a possible solution, since the matter is then in an almost fully ionized, plasma state which can be confined with a magnetic field. As a consequence of the Poincaré Hopf theorem [1], the only topology in three dimensions which has non-vanishing continuous tangent vector field is the torus. Accordingly, in order to prevent end losses, the most successful magnetic confinement fusion devices, the stellarator [2] and the tokamak [3] have toroidal magnetic geometry; i.e. their magnetic field lines trace out nested toroidal surfaces. The tokamak is axisymmetric and the twist of its magnetic field, which is necessary for magnetohydrodynamic stability purposes, is maintained by a current driven inductively in the plasma. According to its relative simplic- 1

14 Chapter 1. Introduction ity, both its theory and technology have developed relatively rapidly, so that the experimentally achieved value of the fusion triple product n i T i τ E, which is the main indicator of fusion performance, has doubled approximately every second year since the mid-1950 s (n i and T i are the ion density and temperature and τ E is the characteristic time of the energy confinement). Since the magnetic field of a stellarator is generated by external coils, its operation does not rely on inductively driven toroidal current, that makes a continuous operation possible, which is clearly desirable in reactors. Also, in contrast to tokamaks, the stellarator lacks current induced instabilities and a hard operational limit on plasma density (Greenwald limit [4]). However, due to the lack of toroidal symmetry, the radial confinement of plasma particles is not guaranteed; its magnetic geometry has to be optimized to avoid drift losses. Especially the confinement of high energy ions is an issue in stellarators. The helical structure of the magnetic field is produced by a complex magnetic coil system. Since the sufficient computational capacity for the optimized design of such complicated, inherently three-dimensional system has been achieved only in the recent years, the stellarator has lagged behind the tokamak concept, in terms of confinement properties and number of experiments. On the road towards controlled thermonuclear fusion it was a notable event when, in 1997, the largest current fusion experiment, JET (Joint European Torus) produced 16 MW of fusion power 65% of the input power [5]. The next step will be an even bigger tokamak experiment, ITER, which is under construction at the present time. Its goal is to demonstrate the technical feasibility of fusion energy production by generating 500 MW fusion power from 50 MW input power [6]. Considerable knowledge on the behavior of fusion plasmas has been accumulated in the recent decades that enabled the design of this experimental fusion reactor. However, most of our predictions regarding the performance of ITER are based on extrapolations using empirical scaling laws [7]. Even today we lack a comprehensive and accurate description of the transport processes in fusion plasmas due to the complexity of the problem. For efficient fusion energy production the energy transport through the magnetic surfaces should be minimized, and at the same time, to maintain the burning plasma, the particle transport has to be kept under control. In addition to the ubiquitous but tolerable level of diffusive collisional transport (which is called neoclassical transport), the major part of the transport is due to convective fluxes associated with 2

15 plasma turbulence. This turbulence is driven by various kinds of smallscale, low-frequency unstable modes, microinstabilities, due to density or temperature gradients [8]. Even hydrodynamic turbulence is a complex unresolved problem, and considering several fluid species coupled through electromagnetic, friction and energy exchange effects, it is not surprising that there is no general theory of plasma turbulence. Understanding and accurately predicting the turbulent transport is one of the most challenging theoretical issues of magnetic confinement fusion. The complexity of the problem nonlinear coupling between the different modes, turbulent cascades through the different spatial scales, nonlinear self-regulation almost makes analytical treatment impossible, although there are methods such as renormalization [9], quasilinear [10] and mixing-length approaches that have been used with limited success. It seems that one has to resort to nonlinear kinetic or fluid simulations to obtain an overall picture of turbulent transport; accordingly, several kinetic and fluid simulation codes have been developed in the recent decades. Nevertheless, to understand the underlying physical mechanisms and to ease the interpretation of simulation results it is rather important to develop reduced models and investigate the properties and different parametric dependences of microinstabilities. The drive of turbulence in the plasma core in a conventional tokamak, is typically dominated by toroidal ion temperature gradient (ITG) modes [11 13] and in certain cases trapped electron (TE) [14 16] modes, but the electron temperature gradient (ETG) mode [17,18], and microtearing modes can also play an important role [19]. In the core, the level of the density and temperature fluctuations is only a few percent of the corresponding equilibrium quantities, while, in the plasma edge and scrape-off layer (SOL) they can be comparable. In the latter, outer plasma regions mainly electrostatic fluid instabilities driven by gradients in pressure, current or resistivity dominate the turbulence. To achieve reactor relevant conditions it is essential to understand, reliably predict and to some extent control turbulent transport. It was discovered that by applying sufficiently high auxiliary heating power, a rapid spontaneous transition to an improved confinement mode the so called high confinement- or H-mode [20] can be obtained, which involves the formation of a transport barrier at the plasma edge, the so called pedestal. In the H-mode pedestal or in artificially induced internal transport barriers [21] the turbulent transport is strongly reduced, in particular the ion energy transport can be on the level of the collisional 3

16 Chapter 1. Introduction transport. Long wavelength neoclassical sheared flows within the flux surfaces play an important role in the suppression of turbulence. For these reasons it is also important to have an accurate knowledge of neoclassical transport. Due to the reduced transport in the edge transport barrier, strong pressure and current gradients can build up, that can destabilize magnetohydrodynamic (MHD) instabilities, leading to abrupt, quasi-periodic, burst-like ejection of a considerable part of the energy stored in the transport barrier region. These instabilities are called ELMs (edge localized modes) and can cause intolerable damage in the plasma limiting elements or at least contribute to their deterioration, but on the other hand they can be useful for helium ash removal [22]. In toroidal geometry density and temperature gradients drive a non-inductive current, the bootstrap current, which plays a crucial role in reactor relevant operation by reducing the need for inductive current drive. This current, which can be considered as one of the most important predictions of neoclassical theory, can destabilize current-driven instabilities, in particular in the plasma edge it contributes to the drive of ELMs [23]. For the deeper understanding of these complex phenomena determining the overall transport of heat and particles, strong interaction between theoretical and experimental work is needed. A routinely applied method for electron density profiles and density fluctuation measurement at the SOL, edge, or outer core regions is beam emission spectroscopy (BES) [24,25]. It is based on the observation of a high energy collimated neutral beam injected into the plasma for heating or purely diagnostic purposes. The photons emitted in the spontaneous de-excitation of collisionally excited beam atoms carry information on the distribution of plasma parameters. The collisions also lead to ionization of the beam atoms, resulting in beam attenuation, which usually restricts the diagnostic to the outer plasma regions. Accordingly, it is well suited for turbulence measurements there, as well as for the investigation of equilibrium or fluctuation driven flows in the edge, together with ELMs, geodesic acoustic modes [26] or other transient phenomena affecting the transport. The remainder of the thesis is organized as follows. In Chapter 2 we introduce basic theoretical concepts that are used in the kinetic modeling of transport phenomena in magnetic confinement fusion devices. In particular we introduce magnetized plasmas and touch upon the particle motion in tokamak geometry before we give an introduction to 4

17 gyro-averaged kinetic theories. In Chapter 3, after a short introduction to collisional transport in general we restrict our attention to neoclassical transport and the relevant kinetic description, the drift kinetic formalism. We give an overview of the different collisionality regimes and sketch the physics of the bootstrap current, and finally discuss the issues of the neoclassical formalism in transport barriers. In Chapter 4 we start with discussing important tools to study microinstabilities and turbulence: the gyrokinetic theory and the ballooning formalism. Then the turbulent and quasilinear fluxes are discussed, before an overview of the characteristics of the two most important microinstabilities, the ion temperature gradient and trapped electron modes is given. A separate section is dedicated to the role of collisions in anomalous transport from both experimental and theoretical perspectives. Chapter 4 is closed by a short discussion of nonlinear simulations. Chapter 5 concerns turbulence and density profile measurements with beam emission spectroscopy. We close by summarizing the included papers in Chapter 6. 5

18 Chapter 1. Introduction 6

19 Chapter 2 Basic concepts in the transport of magnetized plasmas In this chapter we provide a brief overview of the physical and theoretical concepts appearing in the field of transport theory of magnetized plasmas to be used in the following parts of the thesis, however we do not intend to reproduce the derivation of basic plasma physics results. Another aim of this section is to delineate the scope and limitations of the models to be presented. 2.1 Magnetized fusion plasmas In the thesis the physics of fusion plasmas is studied. Plasma is a partially or fully ionized matter. We focus on fully ionized plasmas that are relevant in the core of fusion devices. The plasma consists of unbounded electrons and ions and in the applications of interest these can be considered as classical (i.e. not quantum) objects, that is a very good approximation for energies ( kev) and densities ( m 3 ) typical in magnetic confinement fusion. A complete definition of plasmas involves also spatial and time scales. The plasma is quasi-neutral, that is, in any macroscopic volume the quantity (n e i Z in i )/n e 1, where n e(i) is the density of electrons (ions) and the sum runs over all the ion species with the charge Z i.this approximate condition can be shown to be valid only for length scales larger than the Debye length λ D = ɛ 0 T e /e 2 n e (where T e is the electron 7

20 Chapter 2. Basic concepts in the transport of magnetized plasmas temperature), and time scales longer than the inverse plasma frequency 1/ω pe = m e ɛ 0 /n e e 2. With other words, the plasma does not allow macroscopic charge separation. However, it does not imply that there can be no electrostatic fields present in plasmas, because extremely small charge separation can give rise to considerable electric fields. The ion composition in fusion reactors is dominated by the fuel ions, namely deuterium and tritium. However, apart from a limited number of D-T experiments, the vast majority of the experimental discharges operate with a single species (mostly deuterium). Accordingly, the theoretical work done in transport theory usually considers a single hydrogenic species, which we refer to as the main species. Other ion species are also present in the plasma preferably in smaller quantities; these impurities can come from the plasma facing components interacting with the plasma, and the helium ash of the fusion reactions also constitutes as impurity. For plasmas with several ion species it is useful to introduce the effective ion charge Z eff = ( i n izi 2 ) /ne, which appears in several contexts in fusion plasma physics (for instance in the field of collisional transport), and is an important measure of the purity of plasmas. The plasmas to be considered are magnetized in the sense that the collision frequency ν a for a species a is much smaller than the cyclotron frequency ν a Ω a (Ω a = e a B/m a with the charge and mass of the species e a and m a, and the magnetic field strength B), and that the typical length scale on which the equilibrium plasma parameters vary, L, is much larger than the ion Larmor radius ρ i L (ρ i = v /Ω i, where v is the magnitude of the velocity component perpendicular to the magnetic field. The scale length of a plasma parameter X is usually defined as L X = ln X 1 ). The smallness of the parameter δ = ρ i /L provides the basis of virtually the whole field of transport theory of fusion plasmas operating with gyro-averaged kinetic equations kept through certain order in δ. 2.2 Magnetic geometry and particle motion In a homogeneous magnetic field the particle motion can be conveniently decomposed to a gyration perpendicular to the magnetic field lines with the cyclotron frequency (Larmor motion), and a free motion along the magnetic field line. Thus the particle trajectory traces out a helix with the position of its guiding center R = r (b v)/ω a,wherer is the actual position and v is the velocity of the particle, b = B/B is the 8

21 2.2. Magnetic geometry and particle motion unit vector in the direction of the magnetic field B. In this simple configuration the velocity of the guiding center Ṙ is v = bv,where the parallel velocity is v = b v. If the magnetic field has a spatial variation and/or there is an electrostatic field present, the motion of the guiding center is more complicated: In the parallel direction the particle can be accelerated by the parallel component of the electric field and it is also affected by the magnetic mirror force acting in the direction opposite to the gradient of the field strength. The magnitude of the mirror force is μ B,where μ = mv 2 /(2B) is the magnetic moment of the gyrating particle with v the magnitude of the perpendicular velocity v = v v. Perpendicularly to the magnetic field the guiding center drifts with a velocity v d = Ṙ = E B B 2 + v2 2Ω b ln B + v2 b κ. (2.1) Ω The first term in the right hand side of Eq. (2.1) is called the E B drift, and it is independent of the charge or mass of the particle species. The direction of the other two terms arising due to inhomogeneities in the magnetic field, called the grad-b drift and the curvature drift respectively, depend on the sign of the particle charge. We introduced the curvature vector of the magnetic field κ = b ( b) inthe curvature drift term which, in the limit of low normalized pressure β = 2μ 0 p/b 2 (μ 0 is the vacuum permeability), can be approximated as κ b ln B to get a similar form to the grad-b drift. Then the magnetic drift velocity v D canbewrittensimplyas v D = 1 ( v 2 ) Ω 2 + v2 b ln B. (2.2) The derivation of the drift velocities (or more generally, the guiding center picture of particle motion) assumes that v d v, whichis posteriorly justified for the cases we study. It implies that the motion of an average particle along the magnetic field line is much faster than its drift across the field lines; this is one of the reasons why the geometry of magnetic field lines plays a crucial role in transport theory. The field lines in toroidally symmetric configurations trace out surfaces, which we call magnetic surfaces or flux surfaces. In plasmas in magnetohydrodynamic equilibrium (which we shall consider), the radial pressure gradient is balanced by the J B force, where J is the plasma current density. It follows from the MHD momentum equation J B = p, 9

22 Chapter 2. Basic concepts in the transport of magnetized plasmas that B p =0=J p, which means that currents flow within the flux surfaces that also coincide with the surfaces of constant pressure. As we will see not just the total kinetic pressure, but the densities and temperatures of each species separately are also approximately constants on flux surfaces. However, fast toroidal rotation of heavy impurity ions and other physical phenomena can cause significant poloidal asymmetries. Figure 2.1: Toroidal geometry. r denotes the radial coordinate, φ and θ are the toroidal and the poloidal angles, respectively, and R 0 is the major radius of the torus. Three nested flux surfaces are indicated, and the thick line represents a magnetic field line in the q = 3 magnetic surface. Due to the importance of the flux surfaces, the usual radial coordinate is chosen to be a flux function, i.e. a quantity which is constant on a flux surface, and monotonously increasing between the magnetic axis (the innermost flux surface degenerated to a line) and the last closed flux surface (LCFS) (the last magnetic surface whose field lines do not go through any plasma facing component). It can be the poloidal flux Ψ, which is apart from a constant multiplier the magnetic flux through a surface between the magnetic axis and an arbitrary line lying in a flux surface encircling the torus once toroidally while not making a poloidal turn. Another convenient choice is the toroidal flux Ψ T,that is the magnetic flux trough a loop lying in a flux surface making one poloidal loop. To get a radial coordinate with length dimension, sometimes r Ψ Ψ T /B 0 is used with some reference field B 0. Since both Ψ and Ψ T are (radially monotonously increasing) flux functions they can be written as functions of each other, such as Ψ T (Ψ). 10

23 2.2. Magnetic geometry and particle motion The dimensionless function q(ψ) = dψ T (Ψ)/dΨ, the so called safety factor, is a measure of the helical twistedness of the field lines; it means in a flux surface average sense the number of toroidal turns needed to encircle the flux surface poloidally when following a magnetic field line. In tokamaks q usually (but not necessarily) monotonically increases radially from a value around 1; its radial logarithmic derivative s = r(d ln q/dr) is called the magnetic shear. In Fig. 2.1 a magnetic field line in the q = 3 flux surface is shown together with a set of toroidal coordinates. In toroidally symmetric configurations the magnetic field can be conveniently expressed as B = I(Ψ) ϕ + ϕ Ψ, (2.3) where the first term is the toroidal component of the field B t with I = RB t, the second term is the poloidal field B p,andϕdenotes the toroidal angle. In tokamaks the ratio of the poloidal and toroidal components of the magnetic field is mostly much larger than unity and can be approximated as B t /B p qr/r, withr and r being the major and minor radii, respectively, and their ratio R/r is called the aspect ratio. From B p /B t 1 it follows that Eq. (2.2) describes a drift dominantly in the vertical direction. Therefore a guiding center moving along the field line drifts off from a given flux surface. For purely toroidal field it would lead to charge separation and the loss of confinement due to the rising E B drift. However for finite q the particles spend half of their times (in, say, the upper half of the torus) diverging from a given flux surface and half of their times (in the lower half) approaching the same flux surface again, thus having no net radial drift on time average. The magnetic field strength decreases with major radius as 1/R, which leads to yet another complication in the motion of the particles. Particles having too high magnetic moment compared to their kinetic energy cannot make a complete poloidal turn as they are reflected back from the higher field strength regions by the magnetic mirror force; these are the trapped particles which constitute a ɛ fraction of the particles, where we introduced the inverse aspect ratio ɛ = r/r. These particles bounce back and forth in the outboard side of the tokamak with a poloidal projection of their guiding center trajectory having a banana shape; hence the often used name for the trapped particle orbits: banana orbits. It is clear that for stationary fields the total energy of a particle mv 2 /2+e a Φ has to be a constant of motion, where Φ is the electro- 11

24 Chapter 2. Basic concepts in the transport of magnetized plasmas static potential. In magnetized plasmas, the magnetic moment of a particle μ is an adiabatic invariant, which means that for slow (spatial and temporal) changes in the magnetic field strength the perpendicular velocity of the particles also changes keeping the quantity μ = mv 2 /(2B) a constant. In perfect toroidally symmetric configurations there is an additional constant of motion, the canonical angular momentum Ψ = RA ϕ mrv ϕ /e a =Ψ mrv ϕ /e a, which follows from that the Lagrangian is independent of the toroidal angle ϕ (we introduced the toroidal components of the velocity and the magnetic vector potential, v ϕ and A ϕ respectively). This condition guarantees that the collisionless orbits are confined (i.e., restricted to a radially bounded domain) in tokamaks. In magnetized plasmas not only Ψ is conserved, but also the canonical angular momentum of the guiding centers Ψ. 2.3 Distribution function and kinetic equations The main objective of the kinetic theory of plasmas is to determine the phase space distribution functions of the different plasma particle species f a (t, r, v), from which taking their appropriate moments and averages the desired macroscopic quantities, such as fluxes, flows, currents, can be calculated. The evolution of the distribution functions is determined from the Vlasov equation which is a local conservation equation for the distribution function: df a dt f a t + v f a + e a (E + v B) f a m a v =0, (2.4) where e a (E + v B)/m a is the acceleration due to the Lorentz force. The electric and magnetic fields appearing in Eq. (2.4) can be separated to macroscopic fields, i.e. fields averaged over several Debye length, and microscopic fields that are strongly fluctuating on spatial scales comparable to or smaller than the Debye length due to the discreteness of the particles. The effect of the latter can be collected to a collision operator C a [f a ]= f a coll which describes the change in the distribution as a result of collisions. The resulting kinetic equation, called Fokker-Planck equation, is formally similar to Eq. (2.4) with two differences; the appearing E and B now represent the macroscopic fields, and the right hand side is equal to C a [f a ]. Not only does the collision operator introduce coupling between the different species (noting that C a [f a ]= j C aj[f a,f j ], where C aj describes collisions between species a 12

25 2.3. Distribution function and kinetic equations and j), but the distributions of all species appear implicitly in Eq. (2.4) as sources to the macroscopic electric and magnetic fields. The Fokker-Planck equation together with the Maxwell s equations provide a complete and self-consistent description of plasmas, however this system of equations applied to realistic problems is intractably complex. Realizing that for the description of collisional or turbulent transport processes it is usually unnecessary to resolve time scales corresponding to the fast ion cyclotron motion, the problem can be significantly simplified. A gyro-phase averaging can be performed on the full kinetic equation making use of the smallness of the gyro-radius compared to the equilibrium scales to obtain the simpler drift kinetic or gyrokinetic equations for the gyro-center distribution. The drift kinetic equation used mainly in the theory of collisional transport considers toroidally symmetric fields and represents the particle as a drifting guiding center with a charge and magnetic dipole moment corresponding to the gyration of the particle. The gyrokinetic equation allows for sharp spatial variations in the perturbed fields and distributions on the scale of the ion or electron gyro radius, thus it is more suited for the description of turbulent transport. Having calculated the distribution function of the different species the most important transport quantities, the flux surface average of the radial particle and energy fluxes (Γ a and Q a, respectively) can be calculated as Γ a Ψ = d 3 vf a v Ψ, (2.5) and Q a Ψ = where denotes the flux surface average. d 3 v m av 2 f a v Ψ, (2.6) 2 13

26 Chapter 2. Basic concepts in the transport of magnetized plasmas 14

27 Chapter 3 Neoclassical transport In toroidally symmetric magnetic configurations, which we consider henceforth, such as in tokamaks, in the absence of collisions and fluctuations in local plasma parameters, the trajectories of plasma particles would remain within a radially bounded domain. This result, following from the conservation of the toroidal canonical momentum of the guiding centers, would mean no net radial transport. However, in reality, collisions can move particles from one unperturbed orbit to another, which, on the long term, leads to a diffusive transport of particles and heat across flux surfaces. Even in a cylindrical magnetized plasma there is a radial transport due to collisions. The collisionless orbits are determined by the Larmor gyration of particles around the magnetic field lines and a free streaming of their guiding centers along the field lines. Collisions, by changing the particle velocity perpendicularly to the magnetic field, can relocate the particle from its gyro-orbit to another. The magnitude of the resulting transport can be estimated with a random-walk argument. The particle (electron or ion) diffusivity D e = D i is proportional to ν ei ρ 2 e = ν ie ρ 2 i, where ν ei (ν ie ) is the electron-ion (ion-electron) collision frequency and ρ e (ρ i ) is the electron (ion) Larmor radius. Only unlike particle collisions lead to particle transport, because in like-particle collisions (i.e. electron-electron or ion-ion collisions) the average guiding center position does not change. In an unlike particle collision the electron step length ( ρ e ) is a square root of electron-to-ion mass ratio m e /m i smallerthantheionsteplength( ρ i ), but this is balanced by that the electron-ion collision frequency is a factor m e /m i larger than the ion-electron collision frequency. Therefore the resulting 15

28 Chapter 3. Neoclassical transport electron and ion fluxes are equal Γ e =Γ i. Consequently, the collisional particle transport does not lead to charge separation; this feature is called ambipolarity. The ambipolar property of the particle transport, which originates in the momentum conservation of the Coulomb collisions, persists in the presence of radial electric fields and even in toroidal geometry (in the lowest order in δ = ρ i /L( 1), where L represents the radial length scale). Since energy can be transferred in like-particle collisions there is no condition for heat transport analogous to ambipolarity. In particular the collisional ion energy transport is dominated by ion-ion collisions and is typically a square root of ion-to-electron mass ratio larger than electron energy transport; in terms of ion and electron energy diffusivities χ i ν ii ρ 2 i, while χ e ν ee ρ 2 e. 3.1 Collisional transport across flux surfaces To understand the origin of the collisional transport fluxes in toroidal geometry it is useful to introduce the fluid equations first. By taking the {1,mv,mv 2 /2} velocity moments of the Fokker-Planck equation df a dt f a t + v f a + e a (E + v B) f a m a v = C a[f a ], (3.1) we obtain the conservation equations for particles, momentum and energy [27]: n a t + (n av a )=0, (3.2) m a n a V a + a Π a = ne(e + V B)+ d 3 vc[f]mv, (3.3) t t ( 3nT 2 + mnv 2 2 ) + Q = ene V + d 3 vc[f] mv2 2, (3.4) where we suppressed the species index, and introduced the fluid velocity V = v f, the temperature T = mv 2 f /3, the momentum flux tensor Π = mnvv f and the energy flux Q = mn v 2 v f /2, and the average over the distribution n f = n 1 a fd 3 v. We can denote the last term on the right hand side of Eq. (3.3) by F a (representing the collisional friction force between different species). It is easy to show using Eq. (2.3) that the flux surface average of the particle flux can be written as 16 Γ a Ψ = R ˆϕ (n a V a B), (3.5)

29 3.2. Drift kinetic equation with the toroidal unit vector ˆϕ. Using this identity and taking the flux surface average of the toroidal projection of Eq. (3.3) then yields Γ a Ψ = 1 R ˆϕ m an a V a t e a + R ˆϕ Π a n a e a RE ϕ RF aϕ, (3.6) where ϕ indices denote the toroidal component of a vector. Assuming that the plasma parameter profiles evolve due to collisional diffusion (D νρ 2 ) the time derivatives should be small t D/L 2 δ 2 ν.from this estimate the first term in the right hand side of Eq. (3.6) is at least δ 2 smaller than the last term, even for flow velocities comparable to the thermal velocity. The second term would not be small if it contained contributions from the diagonal of the pressure tensor (i.e. the scalar pressure), but the pressure gradient does not appear due to axisymmetry (ˆϕ p = 0). For similar reason ( ˆϕ Φ = 0) in the third term (the so called Ware-pinch) only the induced electric field appears. The last term arising due to friction between the different species is responsible for the collisional transport. The transport caused by the perpendicular friction is called the classical transport, while the so called neoclassical transport [28] which usually dominates in large aspect ratio tokamaks is due to the combined effect of the parallel induced electric field E (ind) and the parallel friction F. The neoclassical particle flux can be written as Fa + n a e a E (ind) Γ a Ψ neo = I e a B 3.2 Drift kinetic equation. (3.7) The neoclassical fluxes can also be constructed kinetically; the particle and energy fluxes can be written in terms of the distribution function and the drift velocity formally as Eqs. (2.5) and (2.6) with the replacement v v d. The distribution function appearing in the expression for the fluxes can be calculated from the drift kinetic equation. To derive this equation, we will follow the approach of Catto et al [29]. Again, we start with the Fokker-Planck equation (3.1), but rewrite it in a more convenient set of velocity space coordinates: the total energy per particle mass E = v 2 /2+e a Φ/m a that is a constant of motion for time independent electrostatic potential Φ and in the absence of induced electric 17

30 Chapter 3. Neoclassical transport fields (we do not consider the relativistic case, when the particle mass is not constant), the magnetic moment per particle mass μ = v 2 /(2B) that is an adiabatic invariant, and the gyro-angle ϕ. In these coordinates the magnitude of the different velocity components are v 2 =2μB and v 2 =2[(E e aφ/m a ) μb]. In an arbitrary set of phase space coordinates {z i } 6 i=1 the kinetic equation can be written as t f + 6 ż k f/( z k )=C[f], (3.8) k=1 where the time derivative is taken at fixed z and the partial derivative with respect to any z k is taken keeping all the other phase space coordinates and the time fixed. The over-dot acting on any quantity Q denotes Q t Q + v Q +(e a /m a )(E + v B) v Q. Accordingly, f a t + ϕ f a ϕ + E f a E + μ f a μ + v f a = C a [f a ]. (3.9) It can easily be shown that, assuming E E, to lowest order in δ the total time derivatives of the velocity space coordinates are E 0 = e a v (E + Φ) m a μ 0 = μ B v B v B v b v + e a m a B v E ϕ 0 = Ω a. (3.10) Since the largest term in Eq. (3.9) is the one describing the Larmor rotation ( ϕ), the lowest order kinetic equation is Ω a f 0 ϕ =0, (3.11) where the distribution function is given as a series f = f 0 + f [f i+1 /f i = O(δ)]. Equation (3.11) implies that the lowest order distribution is gyro-phase independent. To next order we get f 1 Ω a ϕ + ϕ f 0 1 ϕ + E f 0 0 E + μ f 0 0 μ + v f 0 = C 0 [f 0 ]. (3.12) The gyro-phase average of this equation provides a constraint on f 0 v b f 0 = C 0 [f 0 ], (3.13) 18

31 3.2. Drift kinetic equation where we used that on gyro-phase average the terms containing ϕ- derivatives of single valued functions should vanish, and E 0 ϕ =0= μ 0 ϕ, furthermore the lowest order term in the gyro averaged velocity is the parallel streaming. Taking the transit average of Eq. (3.13) over a full bounce period for trapped particles and over a complete poloidal circuit for the passing particles annihilates the left hand side of the equation yielding dθ dτc[f 0 ]= v b θ C[f 0]=0, where τ denotes time. This constraint has the solution ( ) 3/2 ( ) 3/2 ma f 0 = η a e m ae/t a ma = n a e m av 2 /(2T a ), 2πT a 2πT a that is a Maxwellian distribution, with η a (r) =n a (r) exp[e a Φ(r)/T (r)]. But the collision operator acting on a Maxwellian should vanish (since the system reached local thermodynamic equilibrium), thus Eq. (3.13) reduces to v b f 0 =0. (3.14) Since v is arbitrary, this new constraint implies that f 0 should be constant along the field lines, which together with the requirement of f 0 being continuous means that the lowest order distribution (and also η a ) should be a flux function f 0 = f 0 (Ψ, E). To calculate the first order distribution f 1 it is more convenient to employ the canonical angular momentum Ψ =Ψ m a Rˆζ v/e a (where ˆζ is the unit vector pointing in the toroidal direction) that is a constant of the motion ( Ψ = 0). Considering Ψ as a phase space variable we can define the distribution ( f (Ψ, E) =η (Ψ ) m a 2πT a (Ψ ) which has a very simple total time derivative ) 3/2 e m ae/t a (Ψ ) f = Ψ f + E Ψ f E + f t = E f E + f t. We can write the distribution function as f = f + h which, inserted in the full kinetic equation (3.9), gives ḣ + E f E + f t = C[f + h]. (3.15) 19

32 Chapter 3. Neoclassical transport Writing the unknown part of the distribution h = h 1 + h ,using that E = e a [ t Φ v ( t A)]/m e is small according to the transport ordering, to lowest order we find h 1 Ω a ϕ = C 0[f 0 ]=C 0 [f M ] 0, thus h 1 is independent of the gyro-angle. Using this fact, the next order equation can be written as h 2 Ω a ϕ + v h 1 + μ h 1 μ + e a v (E + Φ) f 0 m a E = C[f + h 1 ]. (3.16) Since the collision operator acting on a Maxwellian is zero, we can make the replacement C[f + h 1 ] C[f f 0 + h 1 ]. Gyro-averaging Eq. (3.16) leads to the drift kinetic equation v b h 1 e [ a f 0 v T b (E + Φ) = C l h 1 Iv f 0 a Ω a Ψ ], (3.17) where we used that the gyro-average of μ vanishes, and approximated f by its Taylor expansion about Ψ to the first order f = f M (E, Ψ) + (Ψ Ψ) f M Ψ + f 0 m a Rˆζ v f 0 E e a Ψ. (3.18) E In Eq. (3.17) we approximated the Coulomb collision operator C with the linearized collision operator C l and used the rotational symmetry of C l to write C l [f f 0 + h 1 ] = C l [ f f 0 + h 1 ]. For ions the second term in the left hand side of Eq. (3.17) arising due to the induced electric field is negligibly small, furthermore usually only the ion-ion collisions need to be taken into account. The electron drift kinetic equation is more complicated as the electrons are collisionally coupled to the ions, and the effect of the induced electric field is not negligible. It can simplify the latter problem if the solution f S of the Spitzer problem C l [f S ]= e T e f 0e v b (E + Φ) is known. The drift kinetic equation is sometimes written in an alternative form. The gyro-average of the first order correction to the lowest order Maxwellian can be written as f 1 = h 1 Iv f 0 Ω a Ψ. E 20

33 3.3. Collisionality regimes Realizing that the radial component of the guiding center drift velocity [Eq. (2.1)] can be conveniently expressed as ( ) v v d Ψ=Iv b E,μ, (3.19) Ω a the drift kinetic equation for f 1 can be easily derived v b E,μ f1 C l f1 = v d Ψ f 0 Ψ + e a f 0 v E T b (E + Φ). (3.20) a 3.3 Collisionality regimes Depending on the relative magnitudes of the effective collision frequency ν eff that is, the typical frequency of the collisional de-trapping of trapped particles and the typical frequencies for the circulating and trapped particles to complete a full orbit (a full poloidal circuit or a banana orbit respectively), we can identify three collisionality regimes. In the low collisionality limit, the so called banana regime that is typical in the high temperature core in the tokamaks, both trapped and circulating particles can complete their orbits before they become detrapped/trapped by collisions. In the high collisionality limit, that is called the Pfirsch-Schlüter (PS) regime, both circulating and trapped particle orbits are frequently interrupted by collisions. For large aspect ratio (ɛ 1) there is a third, intermediate collisionality regime, the plateau regime, where the circulating particles are weakly collisional, like in the banana regime, but the trapped and barely passing particle orbits are frequently interrupted by collisions (due to the lower parallel velocity of these particles). Mathematically, in the PS regime the typical mean free path v T /ν is much smaller than the connection length qr, thusνqr/v T 1, while in the banana regime the effective collision frequency (the frequency of the collisional de-trapping of trapped particles ν eff ν/ɛ) is much smaller than the bounce frequency ω b ɛv T /(qr), accordingly νqr/v T ɛ 3/2. Finally, the plateau regime that occupies the range between these two is characterized by ɛ 3/2 νqr/v T 1. In the banana regime the main part of the radial collisional transport is caused by the trapped particles. Due to the magnetic drifts these particles have radially extended guiding center orbits. Using the conservation of the canonical angular momentum of the guiding centers Ψ the radial width of the banana orbits can be easily estimated as Δ b ɛρ p, 21

34 Chapter 3. Neoclassical transport where ρ p = ρb t /B p ( ρ) denotes the poloidal Larmor radius. Collisions, by interrupting and de-trapping particles at different points of their banana orbits can therefore cause a radial displacement of particles with a step length comparable to Δ b, and in the long term they lead to a diffusive transport. The diffusivity in the banana regime can be estimated from a random walk picture as D b ɛν eff Δ 2 b νρ2 q 2 /ɛ 3/2,wherethe ɛ factor comes from the trapped fraction. This diffusivity significantly exceeds the classical diffusivity D cl νρ 2 discussed in the beginning of Chapter 3, as q is typically higher than unity (except really close to the magnetic axis) and ɛ is smaller than unity. To demonstrate that the contribution of the circulating particles is lower than that of the banana particles we note that a typical circulating particle gets trapped on the time scale of the collision frequency ν ( ν eff ) and the radial extent of its orbit is smaller by a factor of ɛ than a typical banana width, which leads to a diffusion coefficient q 2 ρ 2 ν D b. In the PS regime the parallel motion of the particles is frequently interrupted by collisions and as they change directions they drift upwards or downwards with the magnetic drift frequency. We can use a random walk model to give an estimate for the diffusivity. The parallel step length is the mean free path λ v T /ν which gives a parallel diffusion coefficient D λ 2 ν vt 2 /ν. The time to go around the torus poloidally in a random walk fashion is therefore Δt (qr) 2 /D. The radial step length is then Δr v d Δt ρv T Δt/R, from which the radial PS diffusion coefficient can be calculated as D PS (Δr) 2 /Δt νρ 2 q 2. Since this collisionality regime is typical in the plasma edge, where the safety factor q is larger than unity, the PS transport also exceeds the classical transport. The diffusion coefficients in both of the above collisionality regimes exhibit a linear dependence on collision frequency. Interestingly for intermediate collisionalities, where the passing particles are collisionless and the trapped ones are collisional (i.e. the plateau regime), the diffusion coefficients become independent of collision frequency, although the transport is caused by collisions. This is rather convenient since the details of the collision process (and the form of the collision operator used) become unimportant. The transport in the plateau regime is caused by a resonance between the transit- and collision frequencies of slowly circulating particles [mathematically: slowly as (v /v) 3 νqr/v 1, but circulating ɛ v /v( =. ξ)]. To give a simple random walk es- 22

35 3.3. Collisionality regimes timate in this region the usual (Δr) 2 /Δt should be multiplied by the fraction of the resonant particles F ξ (νqr/v) 1/3. The step time is the inverse of the effective collision frequency ν eff = ν/ξ 2 and the step size is Δr v d /ν eff, which leads to diffusivities with the magnitude D p qρ 2 v T /R. Not surprisingly, this estimate is obtained if we replace νqr/v T by ɛ 3/2 in the banana diffusivity estimate or by 1 in the PS estimate. The collisionality dependence of the neoclassical diffusivity in the banana, plateau and P-S collisionality regimes is illustrated in Fig Figure 3.1: Classical (dashed) and neoclassical (solid) diffusivities as functions of the collisionality in large aspect ratio (ɛ 1). For increasing ɛ the neoclassical diffusivity curve becomes more and more smooth and the plateau region shrinks until it completely disappears for ɛ O(1). To demonstrate how transport arises in the plateau regime we solve the ion drift kinetic equation, which, after introducing the distribution H i = h 1i Iv f 0i Ω i T i ( mi v 2 2T i 5 2 ) ln Ti Ψ, and considering a large aspect ratio circular cross section plasma, can be written in the form of ξ H i θ + ν effqr H i = Q i sin θ, (3.21) v where Q i is a source term proportional to the ion temperature gradient, and the collision is modeled with a simple Krook operator C[H i ] ν eff H i. The radial fluxes are calculated as the flux surface average of certain velocity moments of fv d ˆr, andweknowthatv d ˆr sin θ, so only that part of the solution H i of Eq. (3.21) which is odd in theta 23

36 Chapter 3. Neoclassical transport needs to be kept. It is easy to show that Δ H i(odd) = Q i ξ 2 sin θ, +Δ2 where Δ = ν eff qr/v 1. In the Δ 0 limit the function Δ/(ξ 2 +Δ 2 ) approaches a Dirac delta function times a constant. Physically it means that mainly those particles contribute to the transport for which ξ = v /v is small enough that it becomes comparable to Δ, that is, the ones having similar effective collision frequencies and transit frequencies. We note that using more sophisticated collision operators would lead to the same result [27]. 3.4 Bootstrap current In a cylindrical straight field line geometry parallel current arises only due to parallel electric field and is determined by the conductivity. This contribution to the total current in toroidal geometry is called the Spitzer current (or ohmic current). However in the toroidal case there are further components of the parallel current. Since the surface area of a flux surface in the outboard side is higher than in the inboard side, and the kinetic pressure is constant on a flux surface, a force pointing towards the outboard side is acting on the plasma column. There are perpendicular currents flowing within the flux surfaces to counteract this effect by their J B force. To make the total current divergence free a parallel return current, the Pfirsch-Schlüter current, arises. This current has opposite directions in the inboard and outboard sides and vanishes on flux surface average. Apart from the ohmic and PS currents there is a further component of the total parallel current that does not vanish on flux surface average, the so called bootstrap current generated by density and temperature gradients. To understand the physical mechanism behind the bootstrap current, first we introduce the concept of diamagnetic flow. A particle orbit deviates from the a given magnetic surface to a certain extent partly because of the finite Larmor radius of the particle and partly due to the drifts of the guiding center. The former effect is present even in cylindrical geometry, and having radial gradients in the density and temperature of the species causes a fluid flow perpendicularly to the field lines [in the direction of sign(e a )b {n, T }, the so called diamagnetic direction]. In a certain spatial point there are particles passing with their 24

37 3.4. Bootstrap current guiding centers being in the higher or the lower density (temperature) region; these two groups of particles have opposite velocity components in the diamagnetic direction. Since in the higher density (temperature) region there are more particles (having higher velocity), there will be more particles moving in the diamagnetic direction than the opposite direction, hence causing a net flow, that turns out to be proportional to the pressure gradient. This is a purely fluid flow, since the particle guiding centers (the average position of particles with regard to a field line) are staying still. In toroidal geometry, the magnetic drift of particles causes even larger deviations from a reference flux surface (for trapped particles comparable to the banana width). The particles are mainly moving parallel to the field lines but due to the drifts they pass higher and lower pressure regions, which, by a process analogous to the one causing the usual diamagnetic flow, generates a parallel flow, again, proportional to the gradients in plasma profiles (see the illustration in Fig. 3.2).? p net parallel flow Figure 3.2: Parallel diamagnetic flow of trapped particle guiding centers (top view). It can be shown that the diamagnetic flow of the passing electrons is higher than that of the trapped ones by a factor of ɛ 1/2, even though the banana width is larger than the deviation of circulating orbits from flux surfaces. Electron-electron collisions conserve momentum, therefore the momentum loss of trapped electrons in collisions with the circulating ones and the momentum loss of circulating particles in collisions with the trapped ones should balance. If the parallel flows would simply 25

38 Chapter 3. Neoclassical transport consist of the diamagnetic flows caused by the guiding center motion, the parallel momentum balance could not be achieved. Accordingly there should be an additional parallel flow of circulating particles, which is interestingly even higher than the diamagnetic flow of these particles. This flow produces the bootstrap current. For radially monotonically decreasing density and temperature profiles (which is almost always the case) the bootstrap current has the same direction as the ohmic current, thus reducing the need of the ohmic current drive which limits the length of the plasma discharges. Therefore, it is considered to be advantageous for reactor relevant operation of tokamaks. However, it can be shown that a tokamak cannot operate on purely bootstrap current because the decreasing gradients towards the magnetic axis would lead to magnetohydrodynamically unstable magnetic configurations. The bootstrap current also plays an important role as a destabilizing factor for certain MHD instabilities. 3.5 Neoclassical transport in H-mode pedestals The deviation of guiding centers from flux surfaces (the orbit width) scales as the poloidal Larmor radius ρ p, although formally smaller than that by a factor of ɛ for trapped and ɛ for circulating particles. However in realistic magnetic geometries at the edge ɛ can be O(1), and the banana width becomes comparable to ρ p. There is experimental evidence showing that the density length scale in an H-mode pedestal can also be as small as the poloidal Larmor radius of ions, which causes two problems. Most of the drift kinetic and gyrokinetic treatments assume that the orbit width of the particles is much smaller than the scale lengths of the background parameters, which might be violated in the pedestal. Another problem is that, assuming sub-sonic toroidal rotation of ions, the lowest order ion pressure balance equation requires the existence of a strong radial electric field dφ dr T i d ln n i, e i dr that significantly modifies the ion orbits. In particular, electrostatic trapping of particles becomes important and the trapped region can be shifted towards the tail of the distribution for sufficiently large radial electric fields, as it is illustrated in Fig In neoclassical theory the fact, that the radial drift term in the kinetic equation [first term in the right hand side of Eq. (3.20)] can be 26

39 3.5. Neoclassical transport in H-mode pedestals v 0 v Figure 3.3: Location of the trapped region in velocity space in the weak radial electric field limit (left) and for finite electric field (right). The lowest order Maxwellian distribution is indicated with a gray-scale density plot. expressed in terms of the parallel motion as in Eq. (3.19), is often used to mathematically simplify the problem since this form has similar structure to the parallel streaming term [the first term in the left hand side of Eq. (3.20)]. This can only be done though, when the poloidal motion of particles is dominated by the parallel streaming (v θ = v b θ in lowest order), which is usually the case. However, in the pedestal the E B drift in the strong radial electrostatic field can have a comparable contribution to the poloidal motion. For this reason, v E B needs to be kept in the same order in the formalism as v, but since v E B θ cannot be expressed in terms of the parallel velocity the above simplifying trick cannot be applied. To make progress, another approach can be used in which the canonical angular momentum (or its gyro-average) replaces the poloidal flux as one of the independent variables, which is convenient because Ψ ( )/( Ψ ) = 0. This method which makes clear the distinction between transit averages and flux surface averages was developed by Kagan and Catto [30]. As a complement to their banana regime calculation, we calculate the corrections to the neoclassical plateau regime transport retaining finite E B departures from flux surfaces in paper C. An overview of these results completed by a zonal flow residual calculation can be found in Ref. [G]. 27

40 Chapter 3. Neoclassical transport 28

41 Chapter 4 Turbulent transport and microinstabilities It was recognized from early fusion experiments that the heat transport across the flux surfaces is so high that it cannot be explained purely by collisional processes, even taking the neoclassical corrections into account. As will become clear in the sequel, turbulent flows of the plasma, referred traditionally as anomalous transport, account for the major part of the particle and heat fluxes. The turbulence is driven by different drift-type microinstabilities that are destabilized by inhomogeneities in the plasma parameters. A rough estimate of the turbulent transport can be obtained using a simple random walk estimate, assuming that the step length Δ is comparable to the ion thermal Larmor radius ρ T = v T /Ω c and the step time τ scales as the inverse of a typical (magnetic or diamagnetic) drift frequency k θ ρ T v T /a v T /a, where we used that the perpendicular wave length 1/k θ is also comparable to the Larmor radius and the scale lengths of plasma parameters scale as the minor radius of the device a. This leads to the gyro-reduced Bohm (or simply gyro-bohm [32]) scaling D Δ 2 /τ ρ 2 v T T a ρ T T a eb. (4.1) The level of transport is determined by the saturation amplitude of the perturbed quantities, and therefore the assumption of different saturation mechanisms leads to different diffusivities. One widely used approach is the mixing-length estimate [33], which balances the drift wave drive due to gradients ( ω α ) against the E B nonlinearity 29

42 Chapter 4. Turbulent transport and microinstabilities leading to a diffusivity D γ/k 2,whereγ is the linear growth rate of the most unstable mode. This is equivalent to a wave breaking picture, where the fluctuation amplitude saturates when the gradients of the perturbed quantities grow to the level of the equilibrium gradients ˆn n k ˆn n/l n. (4.2) Assuming a Boltzmann relation between the density and potential perturbations, (ˆn and ˆφ respectively), this implies e ˆφ/T (k L n ) Gyrokinetic description For investigating microinstabilities and for quantitative calculation of the turbulent fluxes one has to compute the distribution function of the different species, which can be done within the framework of gyrokinetic description of plasmas. This gyro-averaged kinetic description allows for short perpendicular wave lengths k on the order of the (ion or electron) Larmor radius and frequencies ω much lower than the ion cyclotron frequency Ω c. In terms of the small parameter δ = ρ/l ρ/a the ordering of the gyrokinetic theory can be written as ˆf a f a e a ˆφ T a e av Ta  T a k k ω Ω ca δ, (4.3) where hat denotes the perturbed quantities. This ordering is often completed by assuming slow variation of the ensemble-averaged quantities / t δ 2, the so called transport ordering. In the recursive method of deriving the gyrokinetic equation [34] the distribution function is separated to perturbed ˆf a and equilibrium parts f a and these together with the perturbed and equilibrium B and E fields are expanded in δ. The gyrokinetic equation is derived from the Fokker-Planck equation [ t + v + e ( a (E + Ê)+v (B + ˆB) m a ) ] (f a + v ˆf a ) = C a [f a + ˆf a ]. (4.4) From the ensemble average of Eq. (4.4) different orders of the drift kinetic equation can be derived, which, in particular, can be used to determine the lowest order equilibrium distribution that is considered to 30

43 4.1. Gyrokinetic description be known in the gyrokinetic equation (and usually is a Maxwellian in a rotating frame of reference). To obtain an equation for the perturbed distribution ˆf a, we subtract the ensemble average of Eq. (4.4) from Eq. (4.4), yielding [ t + v + e a m a (E + v B) = e a m a (Ê + v ˆB ) (f a + ˆf a ) v v ] ˆf a + C a C a ens D a, (4.5) where we introduced the fluctuation-particle interaction operator D a = (e a /m a ) (Ê + v ˆB) v ˆfa ens. The gyro-average of Eq. (4.5) to first order provides a constraint for ˆf a1 which should then have the form ˆf a1 (x) = e a ˆΦ(x) T a + H a (X), where the first term in the right hand side is called the adiabatic (or Boltzmann) part of the perturbed distribution and the second term is the non adiabatic part. The gyro-average of the next order equation leads to the gyrokinetic equation, which, in axisymmetric toroidal configuration, if the 0 th order toroidal rotation can be neglected, can be written as [34, 35] h a t +(v b + v d ) H a + ˆv d h a C a [H a ]= ˆv d f a0, (4.6) where v d is given in Eq. (2.1), and we introduced the perturbed drift velocity ˆv d = b Û/B and h a(x) =H a (X) e a f a0 Û(X)/T a,withthe following gyro-averaged ϕ quantity Û(X) = ˆΦ(X + ρ) v Â(X + ρ). (4.7) ϕ To allow for sharp variations perpendicularly to the field lines, the perturbed fields Ŷ are expressed through the eikonal (or WKB) approximation Ŷ (x) =Y (X)e ik x, (4.8) where Y and k are spatially slowly varying functions. The gyroaverage is Ŷ ϕ(x) =e ik X Y e ik ρ ϕ. (4.9) 31

44 Chapter 4. Turbulent transport and microinstabilities The average can easily be evaluated in terms of the Bessel function of the first kind J n,usingthat J n (z) = 1 dγe inγ+iz sin γ. (4.10) 2π Thus we have e ik ρ ϕ = J 0 (k ρ), v e ik ρ ϕ = iv J 1 (k ρ)(k b)/k, (4.11) ρe ik ρ ϕ = iρj 1 (k ρ)k /k. Using these we find that ] Û(X) =J 0 (k ρ a ) [ˆΦ(X) v  (X) [J 0(k ρ a )+J 2 (k ρ a )] v2 Ω ca ˆB (X). (4.12) In linear gyrokinetic calculations the nonlinearity in Eq. (4.6) represented by the ˆv d h a term is neglected. But even in the linear case the equation is a partial differential equation for which analytical solution can only be found in the simplest geometry and using further approximations. Usually the linear gyrokinetic equation is solved numerically either by looking for the asymptotic behavior of the time dependent coupled gyrokinetic-maxwell system (this is the initial value solver approach), or by constructing a linear matrix equation from the discretized, Laplace-transformed problem and solving for its eigenmodes (this is the eigenvalue solver approach [35]). The advantage of the latter method is that it can find sub-dominant modes, while the initial value solver methods converge to the most unstable eigenmode. 4.2 Ballooning formalism The turbulent fluctuations in plasmas are typically highly elongated along magnetic field lines (i.e. their parallel wave length 1/k is comparable to the connection length qr), but they have a short perpendicular scale (k ρ 1). The seemingly simplest way to represent such an elementary perturbation would be to write it as a flute-like mode: e i(mχ nϕ), χ and ϕ are the poloidal and toroidal coordinates respectively, and m = nq with the integers m and n. However, this form turns 32

45 4.2. Ballooning formalism out to be useful only on a rational surface (i.e. a surface with rational q) and it is incompatible with the periodicity conditions in χ and ϕ as soon as q is irrational, that is, at any finite distance from the rational surface for finite magnetic shear. The linear mode structure of microinstabilities is not localized around rational surfaces but rather it is radially extended over several rational surfaces (which are, in this case, close to each other due to the high toroidal mode number). Thus, instead of employing a flute-like expansion of the modes, it is more convenient to consider the problem in the ballooning representation which is appropriate for the description of mode structures characterized by short perpendicular and long parallel wavelengths when the magnetic shear is finite [36]. After separating the time dependence e iωt, using an eikonal representation, the n th toroidal harmonic of the perturbed field Ŷ can be expressed as Ŷ n (r, χ, ϕ) =ŷ n (r, χ)e in[ϕ q(r)χ], which can further be written as Ŷ n (r, χ, ϕ) = θ 0 j= Ŷ B,n (χ +2πj, θ 0 )e in[ϕ q(r)(χ+2πj+θ 0)], (4.13) where the ballooning function ŶB,n depending on the extended poloidal angle θ = χ+2πj R has been introduced together with the ballooning angle θ 0, which acts as linear eigenmode label [37]. Physically θ 0 is the poloidal angle where the wave fronts are perpendicular to the flux surfaces. The originally two-dimensional problem for ŷ n (r, χ) witha periodicity condition in χ now reduces to a series of one dimensional calculations for ŶB,n(θ, θ 0 ) with the much simpler condition ŶB,n( θ,θ 0 ) 0. In the limiting case of ρ 0, keeping N terms of the θ 0 series, so that θ 0 = {2πl/N mod 2π} N l=1, gives radially NΔ periodic eigenmode solutions, where Δ = (nq ) 1 is the distance between the adjacent rational surfaces. The use of this expansion becomes apparent if we note that the most unstable mode can usually be calculated by considering only the θ 0 =0term. We would like to emphasize that the elementary modes in ballooning representation are radially periodic. Strictly speaking this would represent the reality only if the different geometry and plasma parameters were constant over the considered radial domain. However, to simplify the mathematical problem we can approximate these parameters (and their derivatives) with their value at a certain radial point within the considered radial domain (we might call this procedure the flattening of the profiles ). This can be justified when the considered radial domain 33

46 Chapter 4. Turbulent transport and microinstabilities is small compared to the size of the device (i.e. in the ρ 0 limit). Radial periodicity and flattening of the profiles are often imposed even in nonlinear simulations to increase computational efficiency, as in this case the differential operators appearing in the gyrokinetic-maxwell system have no explicit dependence on radius and the binormal coordinate, thus the perturbed quantities can be Taylor-expanded in the perpendicular domain. For a low-β, circular cross section, axisymmetric, large aspect ratio equilibrium the linearized gyrokinetic equation for the nonadiabatic part of the distribution g a reads in the ballooning representation as [39] v qr θg a i (ω ω Da ) g a C[g a ]= i e af a0 ( ) ω ω T T a φj0 (z a ), (4.14) a where g = ĝ B,n and φ = ˆΦ B,n in the notation of Eq. (4.13). Here, only purely electrostatic perturbations are considered {Â, ˆB } = 0, and θ 0 = 0 is chosen. The time derivative is expressed in terms of wave frequency t iω. Conventionally ω a T [ ( = ω a 1+ x 2 a 2) 3 ] Lna /L Ta, where ω a is the diamagnetic frequency, x a is the velocity normalized to the thermal speed, L na = [ r (ln n a )] 1 and L Ta = [ r (ln T a )] 1 are the density and temperature scale lengths, ω Da = k θ (v 2 /2+v2 )(cos θ + sθ sin θ)/(ω a R) is the magnetic drift frequency (without the finite beta correction). The argument of the Bessel function being responsible for the finite Larmor radius effects is z a = k θ v a 1+s 2 θ 2 /Ω a. The equilibrium distribution f a0 is taken to be Maxwellian. 4.3 Particle and heat fluxes To illustrate how turbulent fluxes arise first we consider only electrostatic perturbations. The potential perturbation ˆΦ corresponds to a perturbed drift velocity ˆv d = b ˆΦ/B producing an ambipolar particle flow. The flux surface average of this flow gives the particle flux Γ a [8], which can be expressed as Γ a = R ˆn aˆv d ˆr ψ, (4.15) where ˆn a is the perturbed density and ˆr is the radial unit vector. In the electrostatic case the drive term in the gyrokinetic equation [the right hand side of Eqs. (4.6) or (4.14)] is proportional to the potential perturbation, therefore the density or temperature perturbations that are different moments of the perturbed distribution function should also have this property. 34

47 4.3. Particle and heat fluxes Using that (b ˆΦ ) ˆr = ik θ ˆΦ, the particle flux can be written as Γ a = k θ T a n a eb eˆφ T a 2 [ ] ˆn a /n a I eˆφ/t a where (ˆn a /n a )/(eˆφ/t a ) is called the density response. finds that the energy flux is Q a = k θ T 2 a n a eb eˆφ T a 2 I [ ˆTa /T a eˆφ/t a ] ψ ψ, (4.16) Similarly one, (4.17) with the temperature response ( ˆT a /T a )/(eˆφ/t a ). It is important to emphasize that as it is clear from Eqs. (4.16) and (4.17) particle (energy) fluxes rise only when the perturbed density (temperature) and the perturbed potential are out of phase. In particular if the non-adiabatic electron response is neglected there is no particle flux in a plasma without impurities. Trivially, the adiabatic part of the distribution does not lead to radial fluxes. The mechanism of the quasilinear fluxes is illustrated in Fig. 4.1, where the potential is color coded and the density is contour-plotted with dotted lines. Particles drift along equipotential contours (indicated by blue arrows) with the E B drift velocity. The mode propagates in the (electron or ion) diamagnetic direction (gray arrow), and the phase shift between the density and potential perturbations leads to that the the maxima of the density perturbations lag behind the maxima of the potential perturbations. And since the density is apparently higher in the outward (upward in the figure) than the inward flow region the resulting imbalance in the flows leads to a net radial particle flux. The same picture holds for temperature perturbations and energy fluxes. The magnitude of the fluxes depends on the amplitude of the potential perturbations which remains undetermined in the solution of the linear problem. The most reliable method to calculate absolute fluxes is to perform nonlinear simulations, where nonlinear saturation mechanisms set the magnitude of the perturbed quantities. A less accurate, but computationally less expensive (and sometimes even analytically tractable) method is the quasilinear approach. In this case the linear problem is solved for different toroidal mode numbers to obtain the linear responses, then the fluxes are calculated using certain estimates for the magnitude of the perturbed potential based on simple mixing-length arguments or 35

48 Chapter 4. Turbulent transport and microinstabilities E B e r E B E B E e Figure 4.1: Schematic picture on the mechanism behind the radial fluxes driven by electrostatic turbulence. The perturbed potential giving rise to E B flows is color density plotted; the perturbed density is contour plotted with dotted lines. experience with nonlinear simulations. In analytical models usually only one representative linear mode is chosen (corresponding to the highest γ or γ/k 2 ). This approach is useful when the cross-phases between the perturbed quantities are approximately preserved as one moves from linear to non-linear simulations (the relevance of linear cross-phases is discussed in Ref. [38]). Not only electrostatic, but magnetic perturbations can also drive particle and energy fluxes. In the notation of Eqs. (4.6) and (4.7), Eq. (4.15) can be written in terms of functions of the guiding center position as Γ a = R d 3 vh a ˆv d ϕ ˆr, (4.18) where H a is determined by the electrostatic gyrokinetic equation and ˆv d ϕ = b ˆΦ ϕ /B. In the electromagnetic case H a is determined by Eq. (4.6) with a source term including the contributions from v Â, and ˆv d ϕ = b Û /B. 4.4 Microinstabilities The generation of fine-scale turbulence in plasmas is believed to be produced by microinstabilities [40, 41], i.e. instabilities which have wavelengths that are comparable to the ion or electron Larmor radii. To 36 ψ

49 4.4. Microinstabilities obtain an overall picture of turbulent transport, and for the calculation of turbulence saturation levels, nonlinear processes have to be taken into account. However it is useful to identify the possible drives and conditions of turbulent processes. Investigation of linear mode characteristics can provide estimates of stability thresholds and parametric dependences of turbulent fluxes. Since the goal of fusion experiments is to sustain enormous temperature gradients (the 4K ev cryostat of a superconducting device is separated from the 10 kev plasma core only by a few meters or less), the plasma is always far from thermodynamic equilibrium. In this non-equilibrium state the available free energy might be transferred to turbulent flows via instabilities. Drift waves are particularly important class of microinstabilities which have often been invoked as the main source of plasma turbulence. Dissipation through, e.g., collisions or kinetic resonances often plays an important role in the de-stabilization of the drift waves. We mainly focus on these type of instabilities classified as dissipative modes, while some others, the reactive ones, do not require dissipation similarly to a wide range of MHD instabilities. A quite important class of instabilities is predominantly electrostatic, although in spherical tokamaks and in the core of conventional tokamaks, where the kinetic pressure normalized to the magnetic pressure β = p/(b 2 /2μ 0 )is not negligibly small, electromagnetic modes might also play role [41]. In the present thesis we restrict our studies to electrostatic microinstabilities. The microinstabilities have spatial scales that are typically much longer than the Debye length, in addition, they are slow instabilities compared to the plasma waves, so that the quasineutrality α e αˆn α can be shown to be a very good approximation. The quasineutrality condition can be used to obtain a dispersion relation from the density responses of the different species. The spatial and frequency scales of the microinstabilities that are thought to be accountable for the turbulence are indicated in Fig. 4.2 together with MHD and cyclotron waves. Clearly, the microinstabilities have much lower frequencies than the cyclotron frequencies (ω ce, ω ci ), which allows for the use of gyro-averaged equations. The wavelengths are times smaller than the size of the system, ranging from 10 times the ion Larmor radius to the electron Larmor radius. In contrast to the MHD waves, this feature makes them somewhat less sensitive to the shaping effects of the geometry, thus analytical calculations often rely 37

50 Chapter 4. Turbulent transport and microinstabilities on the framework of a circular cross section, large aspect ratio model. The ion temperature gradient (ITG) mode driven by ion magnetic drift and transit resonances, which is recognized as the most important drive of turbulence (to be discussed in detail in Sec [11 13]), is characterized by k θ ρ < i 1, but since the turbulent energy flux spectra usually peak at lower wave numbers (k θ ρ i ), an expansion in the FLR parameter might be appropriate. The trapped electron mode (TE or TEM, see Sec [14 16]), appearing in the same wave number range k θ ρ < i 1, is destabilized by electron magnetic drift resonances or collisional dissipation. The TE and ITG modes have a frequency range between the ion and electron bounce frequencies (ω bi, ω be ) which are comparable with the corresponding transit frequencies (ω ti, ω te ). This means that the trapped electrons bounce several times during a wave period; thus usually a bounce averaged electron gyrokinetic equation is used, in which the parallel streaming term (v b ) is annihilated by the averaging operation. Trapping effects become important when the mode frequency is comparable to or lower than the bounce frequency, since then the particles are moving quickly enough along the field lines to sample the whole toroidal geometry during a mode period. Whereas for electron temperature gradient (ETG) and ITG modes the frequencies are higher than the electron/ion bounce frequencies, respectively. The corresponding trapped particle modes, namely, the trapped electron mode and the trapped ion mode (TIM) [42], have frequencies that are lower than or comparable to the bounce frequencies, as it can be seen in Fig Since the bounce frequency of trapped ions is usually smaller than the ITG mode frequency, the ion trapping is often neglected in ITG studies [43]. As an illustration of the drift wave phenomenon, from the quasineutrality condition one can derive the simplest possible electrostatic drift wave by assuming adiabatic electron response, and deriving the ion response from Eq. (4.14), neglecting the FLR effects (J 0 (z i 0) 1) and all the terms on the left hand side, except iωg i. This wave has the frequency ω = ω e. It is marginally stable since I(ω) = 0, and it propagates on the flux surface in the electron diamagnetic direction. Since the magnetic curvature is neglected, ω Di = 0, this mode does not rely on the toroidal geometry: it is a slab mode. The first term of Eq. (4.14) coming from the compressibility-like v b term of Eq. (4.6) would give sound wave propagation along the field lines. Neglecting this term would mean that the ion inertia is assumed to be infinite, which is 38

51 4.4. Microinstabilities Figure 4.2: Typical spatial and temporal scales of different microinstabilities. TIM trapped ion mode, TEM trapped electron mode, ITG (η i ) ion temperature gradient mode, ETG (η i ) electron temperature gradient mode, CDBM current diffusive ballooning mode, δ p skin-depth (mode); (D) dissipative, (C) collisionless, ES/EM electrostatic/electromagnetic. [Source: J. Plasma Fusion Res. 76, 1280 (2000)] justified if the parallel phase velocity of the wave is much higher than the ion thermal velocity and the magnetic field is not strongly sheared. In the toroidal picture, this term might be neglected if the considered frequency range is much higher than the transit or bounce frequency of ions. Since ρ e ρ i, in the description of ion modes (k θ ρ i 1) the electron FLR effects can always be neglected (drift kinetic electrons are considered). In a local analysis of the ion response, taking collisions into account by a simple energy-dependent Krook model C[g i ] = (ν/x 3 )g i, and replacing the parallel compressibility term by k v, the ion gyrokinetic equation reduces to an algebraic equation with the solution g i = ef [ ( i0 ω ω i x2) ] η i T i ω k v ω DT (x 2 J 0(z i )φ, (4.19) /2+x2 )+iν/x3 where x = v/v Ti and ω DT = ω Di vti 2 /(v2 /2+v2 ). We have introduced 39

52 Chapter 4. Turbulent transport and microinstabilities η i = L n /L T, which is a crucial parameter in ITG theory [41], and we have taken the strongly ballooning limit θ 0. The ion density response appearing in the dispersion relation is the velocity integral of this expression transformed back from guiding center to real space [by a multiplication with J 0 (z i )]. The integral contains poles coming from the resonances in the denominator. These terms - the transit, the magnetic drift, and collisional resonances can destabilize the mode. Depending on whether the first or the second of these resonances dominate the resulting instability is called slab or toroidal ITG mode, respectively. We note that in tokamak core plasmas the toroidal mode is the dominant. If all resonances and FLR effects are neglected (as in the previous example) the (3/2)η i and x 2 η i terms in the numerator cancel out in the velocity integration and the mode is not affected by temperature gradients. In general, the collision term contains a differential operator in velocity space and the parallel ion dynamics term makes the problem a differential equation in real space. Moreover, considering the full dispersion relation with a bounce-averaged electron response term, one obtains an integro-differential equation in phase space, which would be intractable analytically without further approximations. One can make use, for example, of the following considerations: If ν e /ω < 1, then ν i/ω is negligibly small. For ω bi ω DT the parallel dynamics term is much smaller than the magnetic drift term k v ω DT. In addition, if the wavelength is comparable to or longer than the ion gyro-radius, the electron finite Larmor radius corrections can be neglected z < i 1 z e 0. Furthermore, one can try to identify the terms that shape the mode structure, and others setting the mode frequency. Additionally, if ω ω be the parallel dynamics dominates the circulating electron response; these electrons can almost freely follow the potential perturbation and therefore have a Boltzmann response. If the drift frequency for thermal velocities, ω DT, is much lower than the mode frequency, the drift resonance is expected to play minor role. Then the expansion of the integrand in the smallness of ω DT /ω the so called non-resonant expansion might be useful, but the validity of this approximation turns out to be limited [44] Ion temperature gradient mode The most important microinstability affecting the ion thermal confinement is the ion temperature gradient mode (ITG or η i -mode), which is a passing particle mode in the k v Ti ω k v Te frequency range. Clas- 40

53 4.4. Microinstabilities sically the mode was investigated assuming adiabatic electron response and the ion response was calculated as (4.19). Depending on whether the mode is destabilized by the k v or the ω Di resonance the mode is called slab- or toroidal ITG. The former mode, which is basically a coupled drift wave/ion acoustic wave in the presence of a radial ion pressure gradient, appears even if we neglect the magnetic curvature. Its typical ( ) 1/3 frequency can be estimated as ω k 2 v2 Ti ω iη i and it propagates in the ion diamagnetic direction [41]. In toroidal geometry the curvature replaces the acoustic wave as the main driving mechanism. The quasi-neutrality condition leads to an eigenfunction problem in the ballooning angle with solutions peaking near θ = 0, i.e., in the bad-curvature region, thus showing a ballooning structure. In this region, the magnetic drift acts to destabilize the mode through the ion temperature gradient [41]. The transit resonance term and the FLR effects also play an important role in shaping the ballooning eigenfunction. This mode also propagates in the ion diamagnetic direction and has a frequency approximately ω (ω i ω Di η i ) 1/2. The physical mechanism behind the toroidal ITG mode is the following. If there is a temperature perturbation in the plasma on the outboard side of the torus, the (mainly vertical) magnetic drift of particles will have different velocities in the lower and higher temperature regions, which leads to a growing density perturbation that is out of phase with the temperature perturbation. The density perturbation generates a potential perturbation, which, in turn, will generate E B flows. The phase of the perturbed flows with regard to the temperature perturbations is such that they convect hot plasma to the already higher temperature spots of the temperature perturbation, leading to the growth of the perturbation amplitude. On the inboard side the relative direction of T is the opposite of B and the magnetic curvature, and the ITG instability is stabilized (thus we often refer to the outboard and inboard sides of the torus as unfavorable and favorable curvature regions). The poloidal angle dependence of the drive and the FLR stabilization shapes the typical ballooning form of the mode structure. An important feature of the ITG mode which appears in both slab and toroidal cases is that the mode is stable below a critical temperature gradient threshold, as illustrated in Fig Within the adiabatic electron approximation and for pure hydrogen plasma, the mode has no unstable roots for η i = 0. Retaining parallel ion dynamics is found to be stabilizing through Landau damping, which thus 41

54 Chapter 4. Turbulent transport and microinstabilities Figure 4.3: Ion heat diffusivity as a function of the logarithmic temperature gradient calculated with different gyro-kinetic and gyro-fluid codes. [Source: Phys Plasmas 7, 969 (2000)] introduces a q dependence, since in tokamaks the parallel wavelength is comparable to the connection length Rq [45]. Introducing a nonadiabatic trapped electron response, a new root, the trapped electron mode appears which can be clearly distinguished from the ITG mode if η i, η e and the trapped electron fraction are high, although there are parameter regimes where the two modes form a single hybrid mode [46], which can be unstable for η i values lower than the classical threshold value (unstable modes with η i = 0 are possible in case of impure plasma as well). For toroidal ITG the relevant parameters are R/L n and R/L T instead of η i and ɛ n = L n /R. We note that the slab mode can be relevant in toroidal geometry as well, when the magnetic drift frequency becomes much smaller than the diamagnetic frequency (when ɛ n is small), which is typical in the edge region, where the density profile is not flat [8]. Turbulent fluctuations can generate zero toroidal and poloidal mode number perturbations which are then not damped by Landau-damping. These, so called zonal-flows [47], appear as predominantly poloidal flows within flux surfaces; the direction of which varies on a radial scale comparable to the ion Larmor radius. Zonal flows, together with neoclassical equilibrium flows, have a strong linear stabilizing effect [48]. In addition, the radial correlation 42

55 4.4. Microinstabilities length of turbulent structures is decreased by the flows, which leads to reduced transport for a given fluctuation level. This double effect is shown in Fig. 4.3 where the ion heat diffusivity curve showing higher threshold corresponds to higher E B shearing rate. This kind of turbulence suppression is recognized to be important for the non-linear self regulation of the plasma and transport barrier formation Trapped electron mode The magnetic field strength in tokamaks decreases from the inboard side of the torus ( high field side or HFS) towards the outboard side ( low field side or LFS) approximately as 1/R. Thus the gyrating particles behaving like small magnetic dipoles following the field lines experience a magnetic mirror force. An O( ɛ) fraction of particles with parallel velocity at the outboard mid-plane lower than v Bmax /B min 1isreflected back from the high-field region, bouncing back and forth. These trapped particles spend most of their time in the bad-curvature region, the LFS, thus the curvature drift has a preferred direction (while this effect averages out for the circulating particles), and the associated local electrostatic fields drive E B drifts giving rise to micro-scale instabilities. The trapped particles cannot follow the electrostatic perturbations freely even if their inertia is negligibly small (mathematically: the v b term vanishes on average over the trapped orbits), therefore they behave non-adiabatically. If collisions are not too frequent to de-trap the trapped particles under a bounce period, various kinds of trapped particle instabilities can arise. On the other hand, the collisional de-trapping can turn nonadiabatic particles to adiabatic [49]. Therefore collisions play a crucial role in the theory of trapped particle instabilities. The collision frequency ν is usually defined by the frequency of π/2 angle scattering. It is however convenient to define a higher, effective collision frequency ν eff = ν/ɛ describing the frequency of collisional de-trapping. The trapped electron mode is one of the most important microinstabilities which can dominate the transport in the presence of internal transport barriers and certainly significantly contributes to the anomalous fluxes in tokamaks [40]. The dissipative TEM is destabilized by the combined effect of the electron temperature gradient and the collisions. The response of the circulating particles is dominated by the parallel dynamics due to the small electron inertia, therefore neglecting the O( ω k v Te ) 1 non-adiabatic circulating electron response can 43

56 Chapter 4. Turbulent transport and microinstabilities be adequate [16] in the electrostatic case. For even lower collisionalities the electron magnetic drift frequency resonance destabilizes the mode, which is then called the collisionless TEM. The maximum growth rate typically occurs at k θ ρ i < 1. For high collisionalities the growth rate γ varies approximately as 1/ν. TheTEM stability boundary shows a strong dependence on collisionality and the FLR parameter [16]. In paper D we study electron cyclotron (EC) heated and ohmic (OH) plasmas from the T10 tokamak, where the transport is found to be mainly driven by trapped electron modes. The qualitatively different scaling with the electron-to-ion temperature ratios in the two cases is due to that the TE mode is driven mainly by electron temperature gradients in the EC case and density gradients in the OH case. Collisional stabilization of the TE modes is also found to be important in these experimental scenarios. 4.5 The role of collisions in turbulent transport In the absence of microinstabilities the particle and heat fluxes across the magnetic surfaces are determined by collisional transport processes, as it was discussed in Sec. 3. Since collisional dissipation affects the non-adiabatic response of the particles it plays an important role in the turbulent transport as well. In gases collisions change the velocity of an atom almost instantaneously, so that its trajectory in phase space is a continuous set of line segments. In plasmas, on the other hand, each particle is in a continuously running Coulomb interaction with a large number of other particles being closer than a few Debye lengths. Since small-angle scattering dominates, the phase-space trajectory of a single particle is a smooth curve, and the collision operator describing the variation of the distribution function due to collisions can be expressed as the Δt 0 limit of ( ) ( ) Δv ΔvΔv C [f(v,t)] = v Δt f + v v : f +..., (4.20) 2Δt where the expectation value is denoted by angle bracket. The first term in Eq. (4.20) is responsible for the collisional drag on the particle while the second term describes diffusion in velocity space. The Δv i and Δv i Δv j quantities can be calculated by a statistical description of binary Coulomb collisions between plasma particles. The 44

57 4.5. The role of collisions in turbulent transport dr/r-like integral over the possible impact parameters do not need to be evaluated over an infinite range, since the particles being outside the Debye sphere do not contribute to the integral and the smallest distance between the colliding particles r min is also finite. So the integration limits are to be cut off at λ D and r min, which leads to the appearance of the Coulomb logarithm parameter ln Λ = ln(λ D /r min ) in the collision frequency. A plasma particle is in a continuously running Coulomb interaction with a large number of other particles, and the small-angle scattering events strongly dominate, so the quantities, such as mean-free path or collision time from the classical picture of collisions are to be reconsidered in this context. The collision time τ is defined as the time which is required for an order unity relative change in the velocity of the particle as a result of the cumulated effect of Coulomb interactions. Then the collision frequency is defined as ν =1/τ. The electron-ion collision frequency depends on electron mass m e, electron temperature T e, ion density n i and ion charge Z in the following manner [27] ν ei e4 n i Z 2 ln Λ ɛ 2 0 m1/2 e Te 3/2. (4.21) Since ν ei is independent of m i, the total electron-ion collision frequency in the presence of several ion species can be conveniently written as Z eff ν ei (Z = 1). Due to the high ion-to-electron mass ratio much higher number of elementary interactions is needed for an ion to significantly change its velocity as a result of collision by electrons than vice versa. Accordingly, the relative magnitudes of the different collision frequencies are ν ee ν ei = ν ii mi /m e = ν ie m i /m e (considering singly charged ions). Making use of Eq. (4.20), the most general collision operator describing binary collisions, the so-called Fokker-Planck operator [50] can be derived to be [51] C ab [ fa (v),f b (v ) ] = e2 ae 2 b ln Λ vk u 2 δ kl u k u l u 3 8πɛ 2 0 m2 a ( fa (v) m v l f b (v ) f b(v ) ) vl f b (v) d 3 v, b m a (4.22) where u = v v is the relative velocity of the colliding particles and the indices a and b refer to the colliding species. In several cases it is 45

58 Chapter 4. Turbulent transport and microinstabilities useful to derive an approximate, model collision operator that is simpler, and therefore less accurate than the one given above, but still models the physical phenomena that are important for the problem. It is always required that the collision operator drives the system towards local thermodynamic equilibrium. In particular, if f a and f b are two Maxwellian distributions with equal temperatures and mean velocities the operator should vanish. Since electrons are much lighter than ions, the dominant process in the electron-ion collisions is pitch-angle scattering, driving a velocity diffusion on a constant-energy surface. This process tends to make the electron distribution isotropic in the ion rest frame. The energy transfer is small due to the high ion-to-electron mass ratio m i /m e, so the electron speed is approximately conserved. Furthermore, if the gyro-angle dependence of the collisions can be neglected the collision operator reduces to the pitch-angle scattering operator C ei ν ei x 3 L ν ei 1 e x 3 e 2 ( ξ 1 ξ 2 ) ξ, (4.23) where ξ =cosθ with the pitch-angle θ. If one wants to investigate the collisional de-trapping of trapped electrons, it is sometimes enough to keep only a pitch-angle scattering model operator of electron-ion collisions since trapping depends only on the cosine of the pitch angle, ξ = v /v. In other cases it is merely sufficient that the collision operator drives the particle distribution towards a Maxwellian, so that it reduces the perturbed part of the distribution. An example of such an operator is the so-called Krook model, C = ν(f a f Ma ), where an energy dependence can be included in the collision frequency ν. For other applications it can be important that the collision model conserves energy and momentum, or that it incorporates other physical effects, such as parallel velocity diffusion. A systematic derivation of model collision operators is given by Hirshman and Sigmar [52]. It was found both experimentally and in simulations that collisions can strongly affect anomalous particle transport [53]. Electron density profiles in tokamak cores are usually not completely flat, although the particle sources are mainly localized in the periphery of the plasma. Neoclassical theory predicts an inward particle transport, due to the Ware pinch [54]. However this effect is often not sufficiently strong to explain the experimentally found density peaking, which implies the existence of an anomalous component of the particle pinch. Experimentally, density 46

59 4.5. The role of collisions in turbulent transport Figure 4.4: Density peaking parameter as a function of effective collisionality, measured in H-mode plasmas of the ASDEX-Upgrade tokamak for three different values of the edge safety factor q 95. [Source: Phys. Rev. Lett. 90, (2003)] peaking is found to increase with decreasing collisionality [53,55 57] for a wide range of plasma parameters, as shown in Fig Since fusion power scales as n 2, this phenomenon is crucial for reactor relevant fusion experiments operating with high temperatures and thus having low collisionalities. Figure 4.5: Normalized electron particle flow as a function of electron collision frequency (ν e is normalized to c s /a). [Source: Phys. Plasmas 12, (2005)] If collisions are neglected, the inward particle fluxes can be explained by gyrokinetic theory [58]. The transport is dominated by ITG driven turbulence in most cases, and ITG modes produce an inward particle flux through magnetic curvature effects and thermodiffusion, as predicted by both linear theory and nonlinear gyrokinetic simulations [44]. In 47

60 Chapter 4. Turbulent transport and microinstabilities fact, the particle flux shows a very strong collisionality dependence for low collision frequencies as illustrated in Fig. 4.5; showing the result of nonlinear gyrokinetic simulations using a pitch-angle scattering collision operator. We note that gyro-fluid simulations predict somewhat higher collisionality for the reversal of particle flows [59]. The collisionality dependence of the particle flux at low electron-ion collision frequencies suggest that the non-adiabatic electron response is affected (note that Γ I(ˆn e )andν i 0). The circulating electrons are expected to have a quite weak non-adiabatic response which is almost independent of collisionality [44]. This conclusion is supported by Fig. 4.5, where the trapped and passing contributions in the particle flux are also indicated. The ν ei -like collisionality dependence can be interpreted with the development of a boundary-layer at the trapped-passing boundary. After discussing the effect of electron-ion collisions on microinstabilities and transport we note that ion-ion collisions also play a role in turbulent transport by providing the main stabilization mechanism of zonal-flows, thus affecting the saturated level of transport. In paper A we investigate the collisionality dependence of the quasilinear particle flux for weakly collisional plasmas using a WKB (Wentzel Kramers Brillouin) solution of the electron gyrokinetic equation, where the electron-ion collisions are modeled by the pitch-angle scattering operator. While in this work the mode frequencies are constant input parameters, in paper B we take collisional effects into account through the dispersion relation as well. In this extended model the density responses are calculated without assuming the magnetic drift frequency to be small. 4.6 Nonlinear simulations and transport analysis As it has been established, linear gyrokinetic simulations consider only one exponentially growing toroidal mode which, in the absence of nonlinear mode coupling mechanisms, never saturates. Accordingly, a linear simulations does not provide the magnitude of the perturbed quantities, and the absolute level of the transport, in contrast to nonlinear simulations. Thus, in order to quantitatively compute turbulent transport one has to resort to nonlinear simulations. There are different approaches to solve the nonlinear gyrokinetic- 48

61 4.6. Nonlinear simulations and transport analysis Maxwell system and different levels of sophistication implemented in nonlinear codes. The delta-f codes separate the lowest order (equilibrium) distribution and solve only for the next order deviation from the equilibrium. The full-f codes solve for the whole distribution function to the same accuracy in ρ. Since there is no full-f formalism to handle collisions, the full-f approach has no significant advantage compared to the delta-f approach. The continuum models evolve the discretized 5-D phase space distribution according to the gyrokinetic equations, while the particle-in-cell codes sample the phase space through a large number of quasi-particles and follow their trajectories under the effect of their self-consistently generated electromagnetic fields. All of these methods are solved in an initial value manner. After the exponential growth of linear modes the fluctuations reach an amplitude where nonlinear mode couplings become important, until the system evolves to a fully developed turbulence, where the initial conditions become unimportant (in a statistical sense). Then the simulation is run for a sufficiently long time in this phase until the statistical properties of the desired quantities, such as radial fluxes, are adequate; see an illustration in Fig Figure 4.6: Results of a nonlinear GYRO simulation for the local parameters of a DIII-D discharge. Left: Time trace of the ion energy flux (given in gyro- Bohm units), Right: poloidal wave number spectrum (at the outboard-midplane) of the ion energy flux, averaged over the half of the total simulation time. Exploiting the elongated nature of the turbulent fluctuations in tokamaks gyrokinetic simulation codes use conveniently chosen field aligned coordinate systems. These are rather similar to the ballooning coordi- 49

62 Chapter 4. Turbulent transport and microinstabilities nates, however they are more fitted to handle the E B nonlinearity and zonal flows. The simulation domain is a flux tube that has a radial and binormal extent that is significantly larger than the correlation length in these directions ( 10ρ i ), and follows the field lines over a few connection lengths (again, depending on the parallel correlation length of the turbulence). In the limit of vanishing ρ (that is relevant to large tokamaks, such as ITER) the plasma parameter profiles and their derivatives can be approximated by constant values over the perpendicular simulation domain. Since the perpendicular domain size is chosen to be larger than the perpendicular correlation length, periodic boundary conditions can be applied to increase numerical efficiency. But sometimes, when profile variations and non-local effects are expected to play a role in the considered problem, the periodic boundary conditions and the profileflattening can be relaxed (these are called global simulations). The existing gyrokinetic simulations do not model the time evolution of the background profiles as these being a higher order effect in the ρ expansion, but rather evaluate the radial transport in a particular instant of time at a certain radial location. To quantitatively calculate the energy and particle fluxes the profiles should be known with quite high accuracy, since the fluxes often exhibit a strong nonlinear dependence on the gradients of plasma parameters [60]; changing the gradients by a small amount can lead to significant differences in the fluxes. When drift-wave turbulence models are validated to experiments, instead of calculating the fixed gradient fluxes, it is sometimes more convenient to calculate how the profiles should look like to produce the experimental level of fluxes (the latter can be calculated from power balance if the energy deposition profiles are given). This can be done by iteration schemes requiring a large number of evaluation of the transport in several radial locations [61]. Even one nonlinear gyrokinetic simulation with appropriate phasespace resolution requires considerable computational resources, therefore it is clear that predictive profile calculations are extraordinarily expensive. This shows the advantage of gyro-fluid and/or quasilinear models (e.g. TGLF [62] and the Weiland model [8]) that although not being as accurate can calculate the turbulent fluxes much faster than nonlinear gyrokinetic codes. In larger devices the perpendicular spatial scale of the turbulence is unaffected by the system size and it is mainly linked to the ion Larmor 50

63 4.6. Nonlinear simulations and transport analysis radius ρ i, while the typical frequencies of k θ ρ i 1 fluctuations scale as v Ti /a. From this picture purely based on dimensional arguments we expect that the diffusivities should follow gyro-bohm scaling, that is χ ρ 2 i v Ti/a. The theoretically predicted gyro-bohm scaling is in conflict with the experimentally usually observed improving confinement with increasing isotope mass. In paper E we study the effect of the primary ion species of differing mass and charge on instabilities and transport through first principles gyrokinetic simulations with GYRO. We also present the transport analysis of three balanced beam injection DIII-D discharges having different main ion species: deuterium, hydrogen and helium. 51

64 Chapter 4. Turbulent transport and microinstabilities 52

65 Chapter 5 Beam emission spectroscopy The progress in the understanding of anomalous transport, such as the development of plasma turbulence models, strongly rely on experimental data measured by various kinds of plasma diagnostic tools. Since in such high temperature systems, the methods based on physical contact of the measuring device and the plasma are quite limited, the diagnostics either passively collect radiation or particles emitted by the plasma (passive diagnostics), or observe the interaction of the plasma with some material/radiation introduced externally (active diagnostics) [63]. Beam emission spectroscopy (BES) is an important, widely used active diagnostic tool of fusion plasmas, which is based on the observation of light emitted by a high energy neutral beam injected into the plasma [64]. The measured intensity distribution corresponding to the spontaneous emission from the highest population excited atomic state, the light profile, provides information on the distribution of plasma parameters affecting the beam evolution. Heating beams (neutral beam injection, NBI) observed tangentially to the magnetic field lines are also used for beam emission spectroscopy [65,66], although in the thesis we focus on beams used only for diagnostic purposes [67 70]. The diagnostic beams have much lower beam current ( ma) than the NBI beams, and due to the attenuation of the beam in the plasma, mainly the outer plasma regions can be probed by them. Since the density of beam atoms is m 3 (which is low compared to m 3 plasma densities), the momentum transfer from the beam is negligibly small, and the quantity of deposited beam material is 53

66 Chapter 5. Beam emission spectroscopy usually too low to noticeably modify the Z eff in the plasma, the method is considered to be non-intrusive [71]. A further advantage of BES is that it is not line integrated, but a well localized measurement of plasma parameters. The electron density, the temperature of plasma particle species (mainly T e ) and the distribution of impurities are the relevant parameters determining the beam evolution [72]. Due to the relatively weak temperature dependence of the reaction rates of alkali elements, they are well suited for electron density measurements. We restrict our studies to the prevailing alkali beam emission spectroscopy. One of the main purposes of alkali BES is the electron density fluctuation measurement [67, 69] (with > 0.5 μs temporal resolution [31]) which provides useful statistical or time-resolved information on turbulence, such as frequency-, wave-number spectra, flow velocities, spatial and temporal correlations, or even snapshots of the turbulence. In fluctuation measurements linear but not local response in emitted intensity to density perturbations is assumed, which depends on the equilibrium density profile. The time resolution of such measurements is limited by the photon statistics which depends on the achievable beam current and the efficiency of the observation. Results of BES fluctuation measurements are shown in Fig. 5.1, where in the left figure the broadband turbulent spectrum and the dramatic reduction of the density fluctuations in an L-H transition is plotted, and in the right figure the frequency spectrum of density fluctuations in different radial positions is shown. An other important application of alkali BES is the electron density profile measurement [64, 68, 72]. This measurement, relying on the knowledge of the derivative of the light profile, requires smooth, time averaged light profiles, which limits its time resolution. The spatial resolution of both measurement modes is limited by the characteristic distance covered by a beam atom under the spontaneous decay of the observed transition. In NBI BES measurements the beam is wide enough to make 2- dimensional turbulence measurements possible for a fixed beam position. By means of scanning the beam, poloidally resolved measurements arealsofeasiblewiththe 1 2 cm wide diagnostic beams [67]. In these measurements, the scanning frequency is higher than the achievable sampling frequency of the profile measurements. In several cases the beam is observed from the same poloidal plane, and a time averaged 54

67 Figure 5.1: Deuterium BES measurement on the DIII-D tokamak. Left: spectrogram showing the evolution of turbulent density fluctuations in an L-H transition (r/a =0.65). Right: Power spectra of fluctuations in different radial positions. [Source: Plasma and Fusion Research 2, S1025 (2007)] signal of the fluctuation measurement is used for calculating the density profile which is, in turn, used as an input to the evaluation of the fluctuation measurements. In this configuration, the density calculation is based on the light profile from a beam which is several times wider than the physical beam width. In spite of that, in many experiments a one-dimensional beam is considered in the density calculations. The evolution of the beam in the plasma is accurately described by the collisional radiative model [73] which considers collisional and spontaneous atomic transitions. Each collisional process is characterized by a rate coefficient defined as R = d 3 vσ( v v B ) v v B f α (v), (5.1) where v B is the velocity of the beam atoms, σ is the cross section of the process and f α is the velocity distribution of the colliding species, considered to be a Maxwellian. The rate coefficients depend parametrically on the beam energy and the temperature T α. In terms of the rate coefficients, the evolution of the occupation densities of the atomic states is described by the rate equations, which read in the rest frame 55

68 Chapter 5. Beam emission spectroscopy of beam atoms d t N k = N k A jk + A kj N j (5.2) + α n α j(<k) j(>k) N k R (α) +k + j( k) R (α) jk + R (α) kj N j j( k), where the different plasma species are indexed by α, andthej and k indices run over the different atomic levels; n denotes particle density, N is the population of an atomic state. A kj is the spontaneous transition frequency and R kj is the rate coefficient corresponding to the j k transition. The ionized state is denoted by +, and the ionized beam atom is considered to be lost from the beam. In a simulation the evolution of a finite number of levels are followed, and the populations of the higher principal quantum number states are neglected. After approximations regarding the impurity content Eq. (5.2) can be written in the more compact form [72] dn dx =[n e(x)a(x)+b] N, (5.3) where the different atomic populations are stored in the vector N(x) and the matrices A(x) andb describe collisional and spontaneous atomic transitions. Observing the ι ϕ transition the emitted light intensity is proportional to N ι JA ϕι /v B,wheretheJ is the current density of the beam. Since in alkali BES measurements one spectral line is observed, the evolution of only one atomic population N ι is known directly. 5.1 Turbulence measurements In turbulence measurements it is assumed that the electron density distribution along the beam can be decomposed to a slowly varying density profile and a fluctuating density n e (x, t) =n e0 (x) +ˆn e (x, t) [74]. The time average of ˆn e (x, t) vanishes, and on the time scale of fluctuations the profile is considered to be static. Here, for simplicity, we consider one dimensional fluctuation measurements and neglect the finite thickness of the beam, although the quantities to be introduced can be generalized to two dimensions. 56

69 5.2. Electron density measurements The measured light profile S can be decomposed accordingly to a static and a fluctuating part S(x, t) =S [n α0 (x),t α0 (x)] + Ŝ(n α,t α ), where S can be determined by solving the rate equations (5.3), and the fluctuation part depends on both the static and fluctuating parts of plasma parameters. However, for alkali beams, the effect of temperature and impurity content fluctuations on Ŝ can be neglected. Then, the fluctuating part of the measured light profile can be written in terms of the density fluctuation transfer function h(x, x )as Ŝ(x, t) = x 0 ˆn e (x,t)h(x, x )dx, (5.4) where the transfer function is considered to be dependent only on the static part of the plasma parameter profiles. (In fact, the value of Ŝ(x, t) depends on the density fluctuations at the retarded time t (x x )/v B, but in the present reasoning, this effect is neglected.) The transfer function is needed in the calculation of density fluctuation cross-correlations from light profile cross-correlations. The transfer function can be easily calculated as h(x, x 0 )=S [ n e0 + δ(x x 0 ) ] S[n e0 ], (5.5) in other words, an elementary fluctuation is superimposed on the density profile at point x 0, the difference appearing in the light profile is monitored. Eq. (5.5) reflects the importance of static density profiles in the evaluation of fluctuation measurements. 5.2 Electron density measurements High spatial and temporal resolution electron density profile measurements [31, 68, 72] in the outer regions of fusion plasmas are of great importance, providing useful information on edge phenomena, such as pedestal formation and ELM activity. Also, as we pointed out, the knowledge of the density profile is also an important input to BES fluctuation measurements. In the direct problem such as in a BES measurement design we are interested in the measured light profile given the plasma parameter profiles along the beam line. It consists of the simulation of beam evolution and the modeling of the observation of the emitted light. Considering an ideal (one dimensional) beam, the beam evolution can be calculated 57

70 Chapter 5. Beam emission spectroscopy by the numerical integration of the rate equation system (5.3) with an appropriate initial condition. In a measurement evaluation the inverse problem is solved, where the electron density is to be calculated from the measured light profile I(x). The classical solution of the inverse problem starts from the rate equation for the initial state ι of the observed transition, which is d x N ι = j [n e (x)a ιj (x)+b ιj ] N j (x). (5.6) Using that the light profile I(x) is proportional to N ι, from Eq. (5.6) the electron density can be expressed as n e (x) = d dx (log I) j j N j N ι B ιj. (5.7) N j N ι A ιj Since the relative populations are not known apriori, Eq. (5.7) has to be solved simultaneously with the direct problem [68]. Note that this method does not require the absolute value of the N ι population, only an arbitrarily normalized light profile. For experimentally relevant plasma densities, there exists a point, where the collisional processes acting to populate and de-populate the ι level equalize, therefore the evolution of this population becomes independent of electron density (the measurement is insensitive in the vicinity of this point). Around this blind point the classical method is replaced by a technique, where the density is calculated as a fraction of integral quantities that are non-vanishing in the vicinity of the singular point [72]. This technique relies on the knowledge of the absolute N ι profile, and the parameter α = N ι /I is found iteratively. Recently, a Bayesian probabilistic method has been developed [31], based on the solution of the direct problem. The measured data are compared to direct calculations, and the most probable density profile is chosen. This approach requires much higher computational capacity, and often the constraint of monotonicity of the n e profile. However, it is more stable than the previous techniques even in the vicinity of the singular point and allows for the evaluation of noisy data, accordingly, higher time resolution profile measurements. At the same time it provides the accuracy of the calculation in each point. In paper F, we point out that neglecting the finite beam width, even for diagnostic beams, might cause non-negligible error in the calculated 58

71 5.2. Electron density measurements density profile regularly the underestimation of the pedestal density. We present a de-convolution based inversion algorithm, which, given the measured light profile, calculates the emission distribution along the beam axis, allowing for the use of conventional one-dimensional density reconstruction methods. 59

72 Chapter 5. Beam emission spectroscopy 60

73 Chapter 6 Summary In the present thesis theoretical and experimental aspects of collisional and turbulent transport in tokamaks are addressed. In the first part of the thesis, the most important electrostatic drift wave instabilities driving the turbulent transport, the ion temperature gradient (ITG) and trapped electron (TE) modes, and the quasilinear fluxes driven by them are studied focusing on the effect of collisions. In paper A, the collisionality dependence of quasilinear particle flux due to ITG and TE modes is investigated analytically. For weakly collisional plasmas, we derive the WKB solution of the trapped electron gyrokinetic equation, where the collisions are modeled by the Lorentz operator. In this model the frequencies and growth rates are considered as input parameters, therefore the dependences on different parameters such as collisionality through the eigenfrequency are neglected, and we use a simple, purely real model ballooning potential. In accordance with previously published gyrokinetic simulation results, we find that, far from marginal stability, the inward flux due to ITG modes caused mainly by magnetic curvature effects and thermodiffusion is reversed as electron collisions are introduced. However, if the plasma is close to marginal stability, collisions might even enhance the inward particle transport. We compare the results calculated by using the Lorentz operator and an energy dependent Krook operator and conclude that the form of the collision operator determines the scaling with collisionality and therefore affects the collisionality threshold where the particle flow reverses. The difference between the two models is larger close to marginal stability. We find that, for low collisionalities, due to the boundary layer development of the non-adiabatic electron 61

74 Chapter 6. Summary distribution function at the trapped-passing boundary, the collisional contribution in the particle flux is proportional to the square root of the collisionality. In paper B, we improved our Collisional Model of Electrostatic Turbulence (COMET) regarding several aspects and focus on the stability of the ITG mode and the ITG-driven quasilinear fluxes. Here, also the ion response is calculated in the long wavelength limit, and the mode frequencies are calculated from the quasineutrality constraint. We introduce a shear dependent imaginary part of the ballooning potential, which we motivate by a self-consistent variational solution of the ballooning eigenfunction problem. This is found to be important for the quantitatively accurate calculation of mode frequencies and fluxes. The improved model, where both the particle and energy fluxes are calculated, does not rely on the non-resonant expansion in magnetic drift frequencies. We find that, although the frequencies and growth rates of ITG modes far from the stability threshold are only weakly sensitive to collisionality, the temperature gradient threshold for stability is significantly affected by electron-ion collisions for high enough logarithmic density gradients. The decrease of collisionality destabilizes the ITG mode driving an inward particle flux, which leads to the steepening of the density profile, in agreement with the trend found in experiments on the collisionality dependence of density peaking. Closed analytical expressions for the electron and ion perturbed density and temperature responses have been derived; and simple, but quite accurate algebraic approximations for these quantities are given. In the next part of the thesis we focus on the collisional transport in transport barriers. In Paper C we calculate the neoclassical plateau regime transport in a tokamak pedestal. In tokamak pedestals with subsonic ion flows the radial scale of plasma profiles can be comparable to the ion poloidal Larmor radius, thereby making the radial electrostatic field so strong that the contribution of the E B drift to the poloidal motion of ions can be comparable to the parallel streaming (mathematically, the normalized electric field U = v E B B/v i B p can be order unity). We calculate the modifications to neoclassical plateau regime adopting a novel kinetic approach allowing for short radial scale lengths and strong electric fields. We find that the ion heat diffusivity is reduced for large values of U, as the resonance causing plateau regime transport is shifted toward the tail of the distribution, but it is enhanced by almost 50% 62

75 if U 1. Moreover, the poloidal ion and impurity flows are modified in the pedestal. The altered poloidal ion flow is most pronounced in the region of the strongest radial electric field where it modifies the friction of the electrons with the ions and can lead to an increase in the bootstrap current, by enhancing the coefficient of the ion temperature gradient term. We show that, unlike the banana regime, orbit squeezing does not affect the plateau regime results. After the general and more analytical considerations we address more specific physics questions through gyrokinetic simulations based on measurements. In paper D we investigate the characteristics of microinstabilities in electron cyclotron heated (ECRH) and ohmic discharges in the T10 tokamak, aiming to find insights into the effect of auxiliary heating on the transport. Results from many different devices have shown that impurity accumulation can be reduced by central ECRH, while in some parameter regions ECRH does not affect the electron or impurity density profiles, or even peaking of these profiles is observed. Trapped electron modes are found to be unstable in both the ohmic (OH) and the electron cyclotron (EC) heated scenarios studied. In the OH case the main drive is from the density gradient and in the EC case from the electron temperature gradient. The growth rates and particle fluxes exhibit qualitatively different scaling with the electron-to-ion temperature ratios in the two cases; in the OH case the electron particle flux decreases with this parameter, while it increases in the EC case. This is mainly due to the fact that the dominant drives and the collisionalities are different. Our linear gyrokinetic simulations indicate that the impurity convective flux is negative in both EC and OH cases, but it is significantly lower in the EC case. Furthermore the impurity diffusion coefficient is lower in that case. As a consequence, the impurity peaking factor is lower in the EC; a trend that is consistent with the observations, however according to the simulations it does not change sign when electron cyclotron heating is applied. A sign change in the peaking factor is, therefore probably due to some additional physical mechanism, such as poloidal asymmetry of the impurity ions, not accounted for in the linear gyrokinetic simulations. In paper E the effect of primary ion species of differing charge and mass specifically, deuterium, hydrogen and helium on instabilities and transport is studied in DIII-D plasmas through gyrokinetic simulations with GYRO. The main motivation of this paper is the isotope scaling problem : the experimentally observed favorable scaling of the 63

76 Chapter 6. Summary energy confinement time with isotope mass is in conflict with the gyro- Bohm scaling. In linear simulations under imposed similarity of the profiles there is an isomorphism between the linear growth rates of hydrogen isotopes, but the growth rates are higher for Z>1main ions due to the appearance of the charge in the Poisson equation. On ion scales the most significant effect of the different electron-to-ion mass ratio appears through collisions stabilizing trapped electron modes. In nonlinear simulations significant favorable deviations from pure gyro- Bohm scaling are found due to electron-to-ion mass ratio effects and collisions. The presence of any non-trace impurity species cannot be neglected in a comprehensive simulation of the transport; including carbon impurity in the simulations caused a dramatic reduction of energy fluxes. The transport in the analyzed deuterium and helium discharges could be well reproduced in gyrokinetic and gyrofluid simulations while the energy transport in the hydrogen discharge was much higher than the gyrokinetic predictions taking neoclassical flows into account. This significant discrepancy is the subject of ongoing investigation and should be a basis of future validation efforts. Finally the thesis touches upon a purely experimental problem; the magnitude and characteristics of the error in alkali beam emission spectroscopy (BES) density profile measurements due to finite beam width are analyzed and a deconvolution based correction algorithm is introduced. If the line of sight is far from tangential to the flux surfaces and the beam width is comparable to the scale length of the light profile, the observation might cause an undesired smoothing of the light profile, resulting in the underestimation of the measured electron density. In paper F, the characteristics and magnitude of this systematic error is studied; a general estimation of the maximal relative error is presented depending on plasma parameters and observation geometry. We demonstrate a deconvolution based correction method by its application in simulated BES measurements of the COMPASS and TEXTOR tokamaks. The method gives a good estimate of the emissivity along the beam line from the measured light profile so that the level of the remaining error in the calculated density after correction is in the order of the accuracy of the density profile calculation algorithm. The method allows the use of the conventional one dimensional density calculation algorithms even for configurations, where the finite beam width is not negligible. 64

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81 REFERENCES [59] C. Angioni, A. P. Peeters, F. Jenko, and T. Dannert, Phys. Plasmas 12, (2005). [60] P. Mantica, D. Strintzi, T. Tala, C. Giroud, T. Johnson, H. Leggate, E. Lerche, T. Loarer, A. G. Peeters, A. Salmi, S. Sharapov, D. Van Eester, P. C. de Vries, L. Zabeo, and K.-D. Zastrow, Phys. Rev. Lett. 102, (2009). [61] J. Candy, C. Holland, R.E. Waltz, M.R. Fahey, and E. Belli, Phys. Plasmas 16, (2009). [62] G. Staebler and J. Kinsey, Phys. Plasmas 17, (2010). [63] I. H. Hutchinson, Principles of Plasma Diagnostics; Cambridge University Press (2002). [64] B. Scweer, Fus. Sci. Technol. 49, 404 (2006). [65] G. R. McKee, C. Fenzi, R. J. Fonck, and M. Jakubowski, Rev. Sci. Instrum. 74, (2003). [66] R. J. Fonck, P. A. Duperrex, and S. F. Paul, Rev. Sci. Instrum. 61, 3487 (1990). [67] S. Zoletnik, G. Petravich, A. Bence, M. Berta, S. Fiedler, K. Mc- Cormick, and J. Schweinzer, Rev. Sci. Instrum. 76, (2005). [68] Z. A. Pietrzyk, P. Breger, and D. D. R. Summers, Plasma Phys. Control. Fusion 35, (1993). [69] S. Zoletnik, M. Anton, M. Endler, S. Fiedler, M. Hirsch, K. Mc- Cormick, and J. Schweinzer, Phys. Plasmas 6, 4239 (1999). [70] R. Brandenburg, J. Schweinzer, S. Fiedler, F. Aumayr, and H. P. Winter, Plasma Phys. Control. Fusion 41, 471 (1999). [71] K. McCormick, S. Fiedler, G. Kocsis, J. Schweinzer, and S. Zoletnik, Fus. Eng. Des , 125 (1997). [72] J. Schweinzer, E. Wolfrum, F. Aumayr, M. Pöckl, H. Winter, and A. Unterreiter, Plasma Phys. Control. Fusion 34, 1173 (1992). [73] R. P. Schorn, E. Hintz, D. Rusbüldt, F. Aumayr, M. Schneider, E. Unterreiter, and H. Winter, Applied Physics B 52, 71 (1991). 69

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83 Included papers A-F 71

84

85 Paper A T. Fülöp, I. Pusztai and P. Helander, Collisionality dependence of the quasilinear flux due to microinstabilities, Phys. Plasmas 15, (2008).

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96 Paper B I. Pusztai, T. Fülöp, J. Candy and R. J. Hastie, Collisional model of quasilinear transport driven by toroidal electrostatic ion temperature gradient modes, Phys. Plasmas 16, (2009).

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98 PHYSICS OF PLASMAS 16, Collisional model of quasilinear transport driven by toroidal electrostatic ion temperature gradient modes I. Pusztai, 1 T. Fülöp, 1 J. Candy, 2 and R. J. Hastie 3 1 Department of Radio and Space Science, Chalmers University of Technology, and Euratom-VR Association, SE Göteborg, Sweden 2 General Atomics, P.O. Box 85608, San Diego, California 92186, USA 3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon OX14 3DB, United Kingdom Received 2 April 2009; accepted 15 June 2009; published online 13 July 2009 The stability of ion temperature gradient ITG modes and the quasilinear fluxes driven by them are analyzed in weakly collisional tokamak plasmas using a semianalytical model based on an approximate solution of the gyrokinetic equation, where collisions are modeled by a Lorentz operator. Although the frequencies and growth rates of ITG modes far from threshold are only very weakly sensitive to the collisionality, the a/l Ti threshold for stability is affected significantly by electron-ion collisions. The decrease in collisionality destabilizes the ITG mode driving an inward particle flux, which leads to the steepening of the density profile. Closed analytical expressions for the electron and ion density and temperature responses have been derived without expansion in the smallness of the magnetic drift frequencies. The results have been compared with gyrokinetic simulations with GYRO and illustrated by showing the scalings of the eigenvalues and quasilinear fluxes with collisionality, temperature scale length, and magnetic shear American Institute of Physics. DOI: / I. INTRODUCTION Turbulent transport in tokamak plasmas is considered to be mainly caused by drift waves destabilized by trapped electrons and ion temperature gradients. 1 4 These microinstabilities and their effect on the transport can be studied by complex nonlinear gyrokinetic codes, for example GYRO. 5 To ease the interpretation of the results of these codes and experimental results it is useful to construct simpler models that can, after careful benchmarking with codes, give various parametric scalings. In particular, the collisionality dependence of the microinstabilities is interesting from both experimental and theoretical points of view. On the experimental side, the evolution of the density profile has been shown to depend on the collisionality On the theoretical side, it has been shown that the transport fluxes are dependent on the choice of the collision operator. 11 Numerical simulations of ion temperature gradient ITG and trapped electron TE modes have shown that collisions may influence the sign and the magnitude of the quasilinear fluxes driven by these instabilities. 12 Without collisions, the quasilinear particle flux driven by ITG modes is usually inward due to curvature and thermodiffusion. Gyrokinetic calculations show that collisions drive an outward flux and the particle flux is expected to change sign for very small collisionalities, much smaller than the collisionality achievable in current tokamak experiments. The choice of the model collision operator affects the collisionality threshold for the reversal of the particle flux. 11 This means that collisionless models or models using the Krook model operator are not adequate to calculate the quasilinear transport fluxes for typical experimental parameters. In this work we develop a collisional model for electrostatic turbulence COMET that can be used to analyze the stability of the ITG modes and to derive analytical expressions for the quasilinear fluxes. Much of the theoretical analysis of the effect of collisions on ITG modes has been based on an energy-dependent Krook operator Here, we model the collisions by a Lorentz operator, which automatically incorporates the increasing importance of pitch-angle scattering near the trapped-passing boundary. 16 We focus on weakly collisional plasmas with large aspect ratio and circular cross section and retain the effect of the magnetic drift nonperturbatively. The collisionality dependence of the particle flux driven by microinstabilities has been studied in Ref. 11 under the assumption that the mode frequency and growth rate are independent of the collisionality. In this work, the model presented in Ref. 11 is extended to include the effect of collisions on the eigenfrequency, growth rate, and stability boundaries of the modes. We show that far from marginal stability the collisionality dependence of the ITG eigenfrequency and growth rate is weak and therefore will have a negligible effect on the particle fluxes. However, we found that the a/l Ti stability threshold is sensitive to the electronion collisions. We found an exact ITG stability boundary in the adiabatic limit which incorporates the shear and the finite Larmor radius FLR parameter dependences. To determine the perturbed electrostatic potential selfconsistently would involve the solution of an integrodifferential equation that is analytically intractable. Therefore, as in Ref. 11, we use a model electrostatic potential, valid in the moderate shear region and motivated by a variational method. The model potential used here is improved compared to the one used in Ref. 11, by including a shear dependent imaginary part of the potential. Assuming this balloon X/2009/167/072305/10/$ , American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see

99 Pusztai et al. Phys. Plasmas 16, ing model potential, closed analytical expressions for the electron and ion density and temperature perturbations are derived, without expansion in the smallness of the magnetic drift. These are used to compute the quasilinear particle and energy fluxes. The results of COMET are benchmarked with numerical gyrokinetic simulations with GYRO and are useful to show the scalings with collisionality, magnetic drift frequency, diamagnetic frequency, and ratio of the density and temperature scale lengths. The remainder of the paper is organized as follows. In Sec. II, the perturbed electron and ion density and temperature responses are calculated. In Sec. III the dispersion relation is presented and the dependence of the stability boundaries on collisionality is studied. In Sec. IV the quasilinear transport fluxes are calculated and scalings of the growth rates, eigenfrequencies, and fluxes with temperature scale length, collisionality, and magnetic shear is discussed and compared with gyrokinetic simulations with GYRO. Finally, the results are summarized in Sec. V. FIG. 1. Color online Electrostatic potential as a function of ballooning angle. Solid black line is the potential used in COMET, green dashed line is the potential calculated by a variational approach in Appendix A, red dotted line is the linear GYRO-result without parallel ion motion and blue dashdotted line is the linear GYRO-result with parallel ion motion. The lower curves are the imaginary part of the potential. The parameters are e = i =3, s=1, q=2, =1/6, and L n /R=1/3. II. PERTURBED ELECTRON AND ION RESPONSES The perturbed electron and ion responses are obtained from the linearized gyrokinetic GK equation, 13 v g a qr i Dag a C a g a = i e af a0 T a T a J 0 z a, where g a is the nonadiabatic part of the perturbed distribution function, is the extended poloidal angle, is the perturbed electrostatic potential, f a0 =n a / vta 3 exp x 2 a is the equilibrium Maxwellian distribution function, x a =v/v Ta is the velocity normalized to the thermal speed v Ta =2T a /m a 1/2, n a, T a, m a, and e a are the density, temperature, mass, and charge of species a, a = k T a /e a BL na is the diamagnetic frequency, T a = a 1+x 2 a 3/2 a, a =L na /L Ta, L na = ln n a /r 1, L Ta = ln T a /r 1, are the density and temperature scale lengths, k is the poloidal wave number, Da = k v 2 /2+v 2 cos +s sin / ca R is the magnetic drift frequency, ca =e a B/m a is the cyclotron frequency, B is the equilibrium magnetic field, q is the safety factor, s=r/qdq/dr is the magnetic shear, r and R are the minor and major radii, J 0 is the Bessel function of order zero, and z a =k v / ca.we consider an axisymmetric, large aspect ratio torus with circular magnetic surfaces. We adopt the usual ordering for the relation of the electron/ion bounce frequencies and the eigenfrequency of the mode bi be and we consider weakly collisional plasmas so that e = e / be 1, where e is the electron-ion collision frequency and =r/r is the inverse aspect ratio. The ion self-collisions and ion-electron collisions are neglected C i g i =0, while the electron-ion collisions are modeled by a pitch-angle scattering operator 1 C e = e v 2 B evl, where e v= T /x e 3, T is the electron-ion collision frequency at the thermal speed, =v /v, =2/m e v 2, and =m e v 2 /2B. Experience from GYRO simulations leads us to adopt the following form for the perturbed electrostatic potential: = cos + if s sin 2 H + H, 2 3 where H is the Heaviside function. The coefficient f s in front of the imaginary part depends on the plasma parameters, e.g., shear, safety factor, density, and temperature gradients. Gyrokinetic simulations with GYRO show that the shear dependence of f s is the most important factor and f s can approximately be written as f s = 0.6s+s 2 0.3s 3. We outline a motivation for the chosen model in Appendix A, where we construct an approximation for the perturbed electrostatic potential in the collisionless limit by a variational method using a trial function similar to the one assumed in Eq. 3. Figure 1 shows that the model electrostatic potential from Eq. 3 is in good agreement with the numerical solution for the potential given by GYRO and the variationally calculated potential calculated in Appendix A. The approximation for the perturbed electrostatic potential breaks down for low and high shear outside the region 0.2s1.7 or near marginal instability, but as we will show, the qualitative features of the transport are captured by our calculations, although for quantitatively accurate results one of course has to resort to numerical simulations. A. Electron response The circulating electrons are assumed to be adiabatic. The nonadiabatic electron distribution can be expanded g e =g e0 +g e1 + in the smallness of / be and the normal- Author complimentary copy. Redistribution subject to AIP license or copyright, see

100 Collisional model of quasilinear transport Phys. Plasmas 16, ized collisionality e, which gives g e0 /=0 in lowest order. The electron GK equation is orbit averaged between the mirror reflection points providing a constraint for g e0, i De g e0 + C e g e0 = ie/t e T e f e0, 4 where is an average over the bounce-orbit of the trapped electrons. Using WKB-analysis to solve the homogeneous equation and then the method of variation of parameters to determine the solution of the inhomogeneous equation it is possible to construct an approximate solution to the orbit averaged GK equation. 11 The homogeneous solution of the electron gyrokinetic equation, obtained by WKB-analysis for weakly collisional plasmas, ˆ e / 0 1, is 1 g hom = u 1/4c 1 sinh z 2 + c 2 cosh z 2, 5 where z= 2u/ˆ 1/4 and =1 B 0 1 /2B 0. In addition, B 0 is the flux-surface averaged magnetic field, 0 =/y is the absolute value of the real part of the eigenfrequency, y=+iˆ, =signr, ˆ is the growth rate normalized to 0, u= iy2 D, and D= D0 / with D0 = k v 2 / ce R is the normalized magnetic drift frequency. In the limit z consistent with the assumption ˆ 1, the inhomogeneous part of the distribution can be simplified to Ŝ g inhom = 22 4 e z 2 /z, 6 D where Ŝ= e 0 /T e 8/31+i4f s /5 T e f e0. In these expressions, we expressed the bounce average of the potential as = 0 E K + i4f s2 1 E 3 K +1, where E and K are complete elliptic integrals. To obtain Eqs. 5 and 6 we approximated the elliptic integrals with their asymptotic limits for small arguments. Since g e0 =0 is regular, we choose c 2 =0, and the boundary condition g e0 =1=0 16,17 gives c 1 = g inhom 1u 1/4 /sinh2 u/ˆ so that the solution for the perturbed trapped-electron distribution is g e0 = Ŝ4 e z 2 /z 22 D Ŝ4 e z 2 / 1/4 /z 2 1/4 2 D sinhz 2 sinhz 2 /, except in a narrow layer close to =0 that has a negligible contribution to the velocity space integrals. The perturbed electron response is proportional to 1 g e0 d v 3 =4 2 v 0 2 Kg e0 d dv0 = 16 2 v 2 dvŝ 0 2 D1 û, 8 where we retained terms only to the lowest order in ˆ 1/2 and approximated 1 0 Kg e0 d2 1 0 g e0 d. The expression in Eq. 8 has been compared with a numerical solution to the 7 electron GK equation resulting in excellent agreement for ˆ up to O1 values. The velocity integral in Eq. 8 can be evaluated in terms of 2 F 0 generalized hypergeometric functions 18 and the perturbed electron density response becomes nˆ e n e e T e =1 2ˆ e 3 e e 2 Dt e 2 ˆ 1 F 5/2 Dt 2ˆ e F 3/ ˆ t 3/2 Dt iy 3 e e F 2 2 7/4 3/2 Dt, 2 9 where =1+4if s /516 0 /3 2, F b a z= 2 F 0 a,b;;z, Dt = D0 /x e 2, ˆ t=ˆx e 3, e = e /, and ˆ a =1 1 3 a /2 a. The perturbed electron temperature can be derived from the nonadiabatic electron distribution function given in Eq. 7 to be Tˆ e e e = 3 2 ˆ e 5 e e T T e 2 2 Dt e 2 ˆ 1 F 7/2 Dt ˆ e F 7/ ˆ t 3/2 Dt iy 7 e e 3/2 F /4 Dt The hypergeometric functions appearing in the perturbed electron and ion responses can be approximated by a simple a algebraic expression F 1 a2 z1 b+ b z c, where the coefficients c and b are given in Appendix B. B. Ion response For the ions we neglect the parallel dynamics by assuming k v Ti. In this limit Eq. 1 can be solved by neglecting the parallel derivative and replacing Di with its weighted flux-surface averaged value Di, where X = Xd/ d. The perturbed ion response becomes nˆ i = e n i T i 1+ d 3 v f i0j 2 0 z i 1 T i / 1 x 2 i Ds e 1+I v, 11 T i where Ds = if ss Di0 121+if s, Di0 = 2k v 2 Ti /3 ci R and we used the constant energy resonance CER approximation for the ion resonance v v 4v 2 +v 2 /3. 13 Author complimentary copy. Redistribution subject to AIP license or copyright, see

101 Pusztai et al. Phys. Plasmas 16, The nonadiabatic part of the ion response can be obtained by the evaluation of the velocity integral of Eq. 11, I v = 2 dx x dx e 0 x2 J 2 0 x 2bs 1 1 x2 3/2 i i 1 x 2 Ds, 12 where we introduced the FLR-parameter b s =b 0 1+s 2 2 with b 0 =k i 2. Also, i =c s / ci is the ion sound Larmor radius, =T e /T i is the electron-to-ion temperature ratio, and c s = Te /m i is the ion sound speed. In order to make further progress analytically, we restrict our analysis to long wavelength perturbations and keep only the linear terms in b 0. This approximation is typically valid for the fastest growing ITG modes k i 0.2. Then I v can be evaluated to obtain the perturbed ion response nˆ i / e = i n i T i + 3 Ds 2 b ˆ i 5 2 i i Ds ˆ i F 1 7/2 Ds, 13 where b=b s =b 0 1+s if s 2 2 3/61+if s is the weighted flux-surface averaged value of the FLR parameter. Note that the expressions for the perturbed electron and ion density in Eqs. 9 and 13 are exact in Da ;no approximation regarding the relative magnitude of Da and has been made. Evaluating an integral similar to Eq. 12 but with the integrand multiplied by x 2 leads to the nonadiabatic perturbed ion temperature response, Tˆ i i e = 3 T T i i i Ds 2 C. Nonresonant expansion b ˆ i 7 2 i i Ds ˆ i F 1 9/2 Ds. 14 In the limit of low normalized magnetic drift frequencies, expanding Eq. 9 around Dt =0 and keeping only the first order terms usually called the nonresonant expansion, the perturbed electron density reduces to the following expression: nˆ e n e e T e =1 21 e + 3 Dt e e 2 3 4i iyˆ t e e y Dt e 4 e. 15 For the ions, expanding Eq. 13 in Ds leads to FIG. 2. Color online F 1 7/2 Ds for y= 1+0.5i. Solid lines: exact. Dotted lines: expansion around Ds =0. Dashed lines: approximation discussed in Appendix B, Eq. B1. nˆ i n i / e T i = b 1 b1+ i i + 3 5b 31+ i 5b1+2 i i Ds However, as we will show, the results based on the expansions in Dt and Ds in Eqs. 15 and 16, respectively, will give large errors compared to the exact solutions in Eqs. 9 and 13. The inability of the nonresonant expansion to reproduce the correct eigenvalues and fluxes has been noted before in Ref. 12. The reason for this is illustrated in Fig. 2 where the difference between the exact and an expanded solution is shown for one of the hypergeometric functions. The solid lines correspond to the real black and imaginary blue parts of the generalized hypergeometric function appearing in the ion response Eq. 13, their expansion to first order around Ds =0 are plotted with dotted lines, while the dashed lines correspond to a simple algebraic approximation discussed in Appendix B. The ITG mode with ˆ =0.5, r = e for R/L n =3, s=1 would correspond to Ds y0.84. D. Summary of analytical formulas Using the approximative formula for the hypergeometric functions from Appendix B, the perturbed electron density, and temperature responses, n a=nˆ at a /n a e and T a=tˆ a/e can be written as n e =1 2 2ˆ e 3 e e Dt ˆ e /2 2 3/5+ 2/5 2 Dt /2 3 4 ˆ t iy 2ˆ e 4/9+ 5/9 2 Dt /2 5/4 3 e e 211/ /21 2 Dt /2 5/2, 17 Author complimentary copy. Redistribution subject to AIP license or copyright, see

102 Collisional model of quasilinear transport Phys. Plasmas 16, /2 T e = 3 5 e e Dt ˆ e /2 2 ˆ e 2 29/14 + 5/14 2 Dt / ˆ t iy ˆ e 11/ /21 2 Dt /2 5/2 7 e e 47/11 + 4/11 2 Dt / The perturbed ion density and temperature responses are n i = i + 3 Ds b 2 ˆ i 5 i i Ds ˆ i 29/14 + 5/14 2 Ds 5/2, T i = i i Ds 2 b 19 ˆ i 7 i i Ds ˆ i 22/3+ 1/9 Ds The formulas, which we summarized above, are the most accurate known for moderate magnetic shear and they can be used to compute the dispersion relation and the quasilinear fluxes as shown in the following chapters. They are useful in showing the scalings with collisionality, magnetic drift frequency, diamagnetic frequency and ratio of the density and temperature scale lengths. III. STABILITY The dispersion relation follows from the quasineutrality condition nˆ i=nˆ e, where the perturbed electron and ion densities are given by Eqs. 9 and 13, respectively, and we take a flux-surface average. The dispersion relation obtained here is valid for both ITG propagating in the ion diamagnetic direction = 1 and TE modes propagating in the electron diamagnetic direction =1, but in this paper, we will focus only on the ITG mode stability and the quasilinear fluxes driven by them. The effect of collisions modeled by a Lorentz operator on the stability of TE modes has been studied before in the steep density and temperature gradient region, 17 where the curvature drift can be neglected. In the limit of large aspect ratio, 0, the trapped part of the perturbed electron density can be neglected and the dispersion relation reduces to the following expression for ITG = 1 modes with adiabatic electrons: 1= i + 3 Ds b 2 ˆ i 5 2 i i Ds ˆ i F 1 7/2 Ds. 21 Using the condition of marginal instability =0, we can derive an approximate stability condition for the ITG modes. We start by noting that the imaginary part of Ds is negligible if 7s 6f s /6+9s+16f 2 s s1 and the imaginary part of b is also negligible when 9f s s 2 / s 2 1 and f 2 s 1, in which case the expression in Eq. 21 is real except for the term containing the function F 2 7/2 Ds that has an imaginary part for all values except Ds =0. Therefore, the condition =0 can only be satisfied if the coefficient of F 2 7/2 Ds vanishes. Using i = e / and Di0 = 2 e n /, where n =L n /R, the 0 for which the coefficient of F 2 7/2 Ds is zero can be shown to be 0 = 2+3s3 i 2 n e 22+3s n 3 i, 22 where we set = 1 for ITG modes. The critical i for stability satisfies the remaining part of Eq. 21, = 1 3 Ds bˆ i i Equation 23 can be rewritten as 3b 1 i +2+3s1 + n =0, so that the ITG stability boundary for adiabatic electrons becomes ic = s n 31 b, 24 and the corresponding critical real frequency of the mode is 0c = b 1 e b s n b For b=0 and s=1, the critical i given in Eq. 24 is similar to what was found previously in the local kinetic limit: 13 ic =41+1/ n /3. In the present model the coefficient of 1+1/ n is 5/3 for s=1, because the flux surface average of the magnetic drift frequency was used instead of its value at =0 as in Ref. 13. If we retain the trapped electron contribution, in the limit of low collisionality e / 0 1 the dispersion relation becomes i + 3 Ds bˆ i i i Ds ˆ i F 1 7/2 Ds 2ˆ e 3 e e 2 Dt =1 1+4if s/532 1+if s ˆ 1 ef 5/2 Dt ˆ t iy 2ˆ e F 3/4 3/2 Dt 3 e F 3/2 Dt 2 2 7/ Author complimentary copy. Redistribution subject to AIP license or copyright, see

103 Pusztai et al. Phys. Plasmas 16, FIG. 3. Color online Stability boundaries of the ITG mode. The upper curves are the critical a/l Ti and the lower ones are the real frequency of the mode normalized to k i c/a. The parameters are e =3, q=2, and =1/6. Left: shear scaling of the stability boundary and real frequency for a/l n =1. The solid black lines are from COMET for adiabatic electrons and the dashdotted blue lines are the results of linear GYRO simulations. Right: a/l n -scaling of the stability boundary and real frequency for s=1. The collisionality increases from the thin to the thick lines: T /k i c/a =0, , 0.025, 0.05, 0.1. The stability boundary for adiabatic electrons is indicated with dotted lines. Figure 3a shows the stability boundaries and real frequencies of the ITG-mode as a function of shear, together with linear GYRO simulations in the collisionless case. The results of COMET are in satisfactory agreement with the results of linear GYRO simulations. Very good agreement with GYRO has been found also in b,, and n scalings. Figure 3b shows the stability boundaries and real frequencies of the ITG mode in the a/l Ti a/l n space for various collisionalities. Clearly, the stability boundaries are significantly affected by the collisionality. The collisionality increases from thin to thick lines and already a small amount of collisionality affects the a/l Ti threshold for high enough a/l n. This means that if we decrease the collisionality, for a constant temperature gradient, much higher density gradient is needed for the stabilization of the ITG mode. IV. QUASILINEAR FLUXES The collisionality dependence of the quasilinear particle fluxes has been studied previously in Ref. 11, but without solving the dispersion relation, and neglecting the effect of the collisionality on these quantities. Here we study the dependence of the transport fluxes including the effect of collisions on the eigenfrequency and we benchmark the results with linear calculations with GYRO. The quasilinear particle flux is ambipolar and is given by e = k p eb e 2 e T e I nˆ e/n e 27 e /T e, where the overbar denotes the flux-surface average of the perturbed quantities and = 0 1+if s /2. The quasilinear energy flux for particle species a is defined by Q a = k p a T a e 2ITˆ eb T a a/t a 28 e /T a. The quasilinear fluxes can be evaluated using the expressions for the perturbed electron and ion density and temperature responses given in Sec. II in Eqs. 9, 11, and 14. Quite accurate approximate results can be obtained also from the formulas listed in Sec. II C, Eqs. 17, 19, and 20, respectively. FIG. 4. Color online a/l Ti scans of the eigenfrequency, growth rate normalized to c s /a, particle flux, and ion and electron energy fluxes normalized to k p e /ebe /T e 2 and k p a T a /ebe /T a 2, respectively. Black solid line is COMET, purple dotted line is with the algebraic approximation to the hypergeometric functions, blue dash-dotted line is the quasilinear GYRO result with ion parallel motion, and green dashed line is the result of the nonresonant expansion. The red dots correspond to nonlinear GYRO simulations. In the following we will present the a/l Ti, collisionality, and shear scalings of the eigenfrequency, growth rate, particle flux and electron and ion energy fluxes, together with quasilinear and nonlinear GYRO results for the following parameter set a/l Te =a/l Ti =3, s=1, q=2, a/r=1/3, a/r=2, and a/l n =1 i.e., the GA standard case 5. Each GYRO nonlinear simulation used a perpendicular domain size of L x / i,l y / i =86,90, fully resolving modes with wave numbers in the range k x i 2.3 and k y i 1.1. The standard 128-point velocity-space grid eight energies, eight pitch angles, and two signs of velocity was used. Electrons were taken to be drift kinetic with mi /m e =60 and the simplified s geometry equilibrium model was used. A. a/l Ti -scaling Figure 4 shows the a/l Ti scalings of the eigenvalues and the fluxes in the collisionless case. The COMET results solid line are compared with quasilinear GYRO blue dash-dotted line and nonlinear GYRO red dots simulation results. For comparison, the results computed by expanding to first order in D / nonresonant expansion are shown with green dashed lines and the results using the algebraic approximations to the hypergeometric functions are shown with purple dotted lines. The frequencies are normalized to c s /a, where a is the minor radius, while the particle and energy fluxes are normalized to k p e /ebe /T e 2 and k p a T a /ebe /T a 2, respectively. The comparisons to the nonlinear simulations are based on the choice of the flux surface averaged perturbed potential amplitude so that e /T e / =6.5, where = i /a, which is consistent with the usual mixing length estimate nˆ e/n e 1/k L n. The agreement between COMET and quasilinear GYRO results is very good. The disagreement with the nonresonant expansion green dashed line is large, but not surprising, since D. Author complimentary copy. Redistribution subject to AIP license or copyright, see

104 Collisional model of quasilinear transport Phys. Plasmas 16, FIG. 5. Color online Collisionality scans of the eigenfrequency, growth rate, particle flux, and ion and electron energy fluxes. Black solid line is COMET, purple dotted line is with the algebraic approximation to the hypergeometric functions, blue dash-dotted line is the quasilinear GYRO result with ion parallel motion, and green dashed line is the result of the nonresonant expansion. The red dots correspond to nonlinear GYRO simulations. FIG. 6. Color online Shear scans of the eigenfrequency, growth rate, particle flux, and ion and electron energy fluxes. Black solid line is COMET, purple dotted line is with the algebraic approximation to the approximate hypergeometric functions, blue dash-dotted line is the quasilinear GYRO result with ion parallel motion, and green dashed line is the result of the nonresonant expansion. The particle fluxes are inward and their absolute values decrease with a/l Ti, but the electron and ion energy fluxes increase with a/l Ti. The disagreement between the full COMET solution and the nonresonant expansion is remarkable for the particle flux, which shows that the nonresonant expansion fails to reproduce both the sign and the magnitude of the particle fluxes. Furthermore, the nonresonant expansion gives incorrect scaling for the electron energy flux as a function of i. B. Collisionality scaling Figure 5 shows the collisionality scaling of the eigenvalues and fluxes. As in Fig. 4 we show the COMET results compared to quasilinear and nonlinear GYRO results together with the nonresonant expansion. The collisionality ei is defined in units of c s /a and is ei =n e e 4 ln / T e 3/2 me, where ln is the Coulomb logarithm. The results show that the eigenfrequency and growth rate of the ITG mode are insensitive to the collisionality, therefore the quasilinear particle flux driven by ITG is almost identical to the one calculated in Ref. 11, where the collisionality dependence of the eigenvalues was neglected. As noted before, the particle flux changes sign from inward to outward at a certain value of the collisionality. The electron and ion energy fluxes depend only weakly on collisions. Again, the agreement between COMET and GYRO results is very good. Also here, the nonresonant solution departs considerably from the full solution, both for the eigenvalues and the fluxes. As noted in previous work, 11 the Krook and Lorentz model operators lead to different results, although for cases far from marginal instability, as it is the case for the parameters used in this paper, they are in qualitative agreement. The collisionality dependence of the electron heat flux differs from the quasilinear GYRO results. The COMET results exhibit almost no dependence on collisions but the quasilinear GYRO results show that collisions drive an outward electron heat flux. This is mainly due to the fact that the assumed ballooning potential is different from the potential calculated by GYRO. In particular, in our model, the part of the potential that is outside the interval of the extended angle, is neglected. This assumption gives reasonable results for the particle flux and the ion heat flux, but not for the electron heat flux, where the contribution of the part outside of, can be large. However, the nonlinear GYRO results show that the collisionality dependence of the electron heat flux is in fact almost negligible, therefore COMET gives qualitatively correct scaling with the collisionality in spite of the fact that it is unable to reproduce the collisionality dependence of the quasilinear GYRO results for the electron heat flux, due to the simplicity of the assumed ballooning potential. C. Shear scaling The validity of the model depends on the assumed electrostatic potential, the form of which is dependent on the shear. Some of the shear dependence is kept by using a shear-dependent factor f s multiplying the imaginary part of the potential, but the width of the real part of the potential is not varied and the dependence of f s on plasma parameters other than shear is neglected. Therefore the model is still too crude to capture all of the shear dependence of the problem. However, as illustrated in Fig. 6, where the shear dependence of the eigenvalues and fluxes are shown in the collisionless case, in the moderate shear region, the COMET results have reasonable agreement with quasilinear GYRO results. V. CONCLUSIONS In this paper we presented a semianalytical collisional model for electrostatic microinstabilities and the quasilinear transport fluxes driven by them. By assuming a ballooning eigenfunction for the electrostatic potential we obtained closed analytical expressions for the perturbed electron/ion density and temperature responses. The expressions contain Author complimentary copy. Redistribution subject to AIP license or copyright, see

105 Pusztai et al. Phys. Plasmas 16, explicitly the dependence on electron-ion collision frequencies and no expansion in the smallness of Da / is used. The collisions are modeled by the Lorentz operator, which gives the proper boundary layer development of the nonadiabatic trapped electron distribution at the trapped-passing boundary as it is shown in Ref. 11. We illustrated that the linear approximation of the hypergeometric functions, the so-called nonresonant expansion, is valid only in a very small vicinity of Da /=0. Therefore it is not appropriate for typical ITG frequencies and it gives incorrect results for the parameter region we studied, where the density profile is not too steep. We introduced a simple algebraic approximation of these functions for Da /=O1 arguments. Our model is semianalytical in the sense that the roots of the dispersion relation are obtained numerically. However, we found an exact ITG stability boundary in Eq. 24 for the 0 and bi / 0 limits, which incorporates not only the L n /R and T e /T i, but also the shear and the FLR-parameter dependences. This stability boundary agrees well with numerical results obtained in GYRO. It has been shown that for a constant density gradient, increasing collisionality leads to a larger a/l Ti threshold for stability. This means that decreasing collisionality gives rise to a destabilization of ITG mode driven turbulence, which in turn may lead to a steepening of the density profile. We benchmarked our model to the recognized, state-ofthe-art gyrokinetic simulation code GYRO and illustrated the agreement on the GA standard case. 5 The results for the eigenfrequencies and growth rates agree well with quasilinear gyrokinetic calculations with GYRO. Our results show that the toroidal ITG frequencies can be calculated accurately neglecting the parallel ion dynamics using a model ballooning potential. However it is important to take the shear dependent imaginary part of the potential into account. We motivated the model potential by a self-consistent variational solution of the ballooning eigenfunction problem. The frequencies and growth rates of the ITG modes far from threshold are not very sensitive to the collisionality, but the quasilinear particle flux changes dramatically for very small collisionalities, in agreement with the results of Ref. 11 and of previously published gyrokinetic simulations, e.g., in Ref. 12. Expressions for the electron and ion particle and energy fluxes have been derived and from comparisons with quasilinear and nonlinear GYRO simulations we concluded that the COMET flux captures the qualitative collisionality dependence and a/l Ti -scalings. The quantitative agreement between COMET and GYRO becomes worse for low and large shears, because the assumed form for the electrostatic potential is not a good approximation in that region. Reliable quantitative predictions for the eigenvalues and fluxes can therefore only be obtained in the moderate shear region. ACKNOWLEDGMENTS This work was funded by the European Communities under Association Contract between EURATOM and Vetenskapsrådet. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors would like to thank P. Helander and H. Nordman for useful discussions of the paper. APPENDIX A: MODEL FOR THE PERTURBED ELECTROSTATIC POTENTIAL In order to motivate the chosen model for the perturbed electrostatic potential, it is convenient to follow a variational approach using a trial function for the perturbed electrostatic potential motivated by GYRO simulations: = 0 1+A cos + B sin Acos 3 H + H. A1 The coefficients A and B are determined by taking 1,cos,sin 2 moments of the integrodifferential equation derived from the quasineutrality condition. A similar method was used by Ref. 19, where a trapped particle instability was considered. The potential was expanded in series in cos and the coefficients were determined by taking moments of the equation resulting from the quasineutrality condition. More recent attempts to obtain a model electrostatic potential see, e.g., Ref. 14 neglected the effect of the trapped particles, motivated by the smallness of the inverse aspect ratio. The neglect of the trapped population simplifies the mathematical problem considerably, since it removes the integral part of the equation. In reality the factor 2 is not very small and the trapped population may have influence on the form of the potential. With the variational method outlined by Ref. 19, the effect of the trapped particle population can be kept, along with the terms resulting from the ion parallel motion. Unfortunately a full analytical solution of the problem is difficult, unless we neglect collisions and expand in the smallness of v Ti /qr, the FLR parameter b and normalized magnetic drift frequency D /. Although these approximations restrict the validity of the model, it is still instructive to study what determines the shape of the ballooning potential. We begin with a simplified version of the problem, by neglecting the D -resonance, ion parallel motion and collisions. Then, the ion response can be obtained by neglecting the term proportional to Ds in Eq. 16 and is nˆ i en e /T e = i + b s 1 i 1+ i, A2 where b s =b 0 1+s 2 2 and the tilde denotes normalization with respect to. Neglecting collisions, the electron response can be obtained directly from Eq. 4, nˆ e = 1 e 1 en e /T e 2 Bd, 1 B A3 where is an average over the bounce-orbit of the trapped electrons and the -integral is over the trapped electron population. Using Eqs. A2 and A3 the quasineutrality condition becomes Author complimentary copy. Redistribution subject to AIP license or copyright, see

106 Collisional model of quasilinear transport Phys. Plasmas 16, nˆ e nˆ i 0= en e 1 e = 1+ i + b s 1 i 1+ i 1 1 e 2 Bd, 1 B and it can be written as an integral equation where q 0 + q 0 1s Bd 1 B =0, A4 A5 q 0 = 1+ i + b 0 1 i 1+ i. A6 1 e For the electrostatic potential we assume the trial function given in Eq. A1 and take moments of the quasineutrality equation, 1 2 nˆ e nˆ i d e 1 cos en e 1 e /T =0, sin 2 A7 to obtain three equations that determine the coefficients of the perturbed potential A, B, and the eigenvalue q 0 that gives the dispersion relation. For the parameters used throughout this paper =1, i = e =3, =1/6, s=1, q=2, and L n /R=1/3, the above simplified approach gives the eigenvalue q 0 =0.46 and the coefficients for the eigenfunction: A=0.75 and B= 1.3. The real part of the potential is similar to the one obtained by GYRO, but due to the neglect of the parallel ion motion and D resonance, the information about the imaginary part of the potential is lost. Note that for the simple ion response given in Eq. A2 the parameters A, B, and q 0 can be calculated without any assumption on the mode frequency. Including parallel ion motion and the effect of the Di resonance to leading order, the ion response is given by nˆ i en e /T e = i b 0 1 i 1+ i + q e b 0 s Di b b b 0s Di, 2 4 A8 where the term proportional to 2 =v 2 Ti / 2 q 2 R 2 represents the contribution from the parallel ion motion and we used the CER approximation. Neglecting electron-ion collisions and the De resonance the electron response is given by Eq. A3. The shape of the ballooning potential and the mode frequency can be self-consistently calculated using the following iteration scheme. By the solution of the three coupled equations resulting from the integrals given in Eq. A5, the parameters A, B, and q 0 can be determined for a given. A new can be obtained from the solution of Eq. A4, which TABLE I. Coefficients in the approximative formulas. 1 F 5/2 1 F 7/2 1 F 9/2 3/2 F 3/4 3/2 F 7/4 3/2 F 11/4 c c 2 5/2 3 5/4 5/2 3 is fed back to the previous system of equations. For our standard parameters it leads to the eigenvalue: q 0 = i, mode frequency = e i, and the coefficients for the perturbed potential are A= i and B = i. The potential corresponding to these values is plotted in Fig. 1 with green dashed line, together with the one computed by GYRO with and without ion parallel motion, and the one we use in the paper. The potential calculated in this way agrees very well with the result of the GYRO simulation. We note that taking into account the effect of the D -resonance to first order and parallel ion motion is important to obtain the correct sign of the factor multiplying the imaginary part of the potential. APPENDIX B: APPROXIMATION OF THE HYPERGEOMETRIC FUNCTIONS IN THE RELEVANT PARAMETER REGIME Since the frequency of the toroidal ITG mode is typically of the order of the magnetic drift frequency, the nonresonant expansion, i.e., the expansion in the smallness of Da /, is not appropriate, as it is illustrated on Fig. 2. In order to obtain simple algebraic expressions for the perturbed density and temperature responses given in Eqs. 9, 13, 11, and 14, respectively we introduce approximations for the generalized hypergeometric functions F 2 a a1 z appearing in these expressions. These approximations are valid for z=o1 arguments and exact in the z 0 limit. In our analysis, we assumed that bi, which gives an upper limit for the argument of the hypergeometric functions, namely, z Da / cannot be much higher than one. Therefore we do not needed the approximation to be valid asymptotically as z. For arguments that are not too large z3, we can approximate the generalized hypergeometric function a F 2 a1 z= 2 F 0 a 1,a 2 ;;z by a F 1 a2 z1 b + b z c, B1 where b=c 2 /4a 1 a 2 and c is a constant that can be found by the minimization of the absolute value of the integral difference between the exact and approximate functions for a finite radius domain of the complex plane. The constant is not a low order rational number in general, but can be approximated with a properly chosen one, still providing a very a good approximation for F 2 a1. The values of c corresponding to the best approximation in the above sense for z3 and their rational approximations are listed in Table I. The average integral difference of the approximative algebraic expression and the exact hypergeometric function on the z3 domain was shown to be below 4% if the constant c is chosen to be a rational number. In Figs. 4 6 the eigenvalues and Author complimentary copy. Redistribution subject to AIP license or copyright, see

107 Pusztai et al. Phys. Plasmas 16, fluxes calculated using the approximative formula B1 with the rational approximations of c are shown with purple dotted lines. The eigenvalues and fluxes calculated with the approximation in Eq. B1 are in excellent agreement with the ones calculated with the exact hypergeometric functions. 1 J. W. Connor and H. R. Wilson, Plasma Phys. Controlled Fusion 36, W. Horton, Rev. Mod. Phys. 71, X. Garbet, P. Mantica, C. Angioni, E. Asp, Y. Baranov, C. Bourdelle, R. Budny, F. Crisanti, G. Cordey, L. Garzotti, N. Kirneva, D. Hogeweij, T. Hoang, F. Imbeaux, E. Joffrin, X. Litaudon, A. Manini, D. C. McDonald, H. Nordman, V. Parail, A. Peeters, F. Ryter, C. Sozzi, M. Valovic, T. Tala, A. Thyagaraja, I. Voitsekhovitch, J. Weiland, H. Weisen, A. Zabolotsky, and the JET EFDA Contributors, Plasma Phys. Controlled Fusion 46, B J. Weiland, Collective Modes in Inhomogeneous Plasma Institute of Physics, Bristol, J. Candy and R. E. Waltz, J. Comput. Phys. 186, C. Angioni, A. G. Peeters, G. V. Pereverzev, F. Ryter, G. Tardini, and the ASDEX Upgrade Team, Phys. Rev. Lett. 90, C. Angioni, A. G. Peeters, G. V. Pereverzev, F. Ryter, and G. Tardini, Phys. Plasmas 10, H. Weisen, A. Zabolotsky, C. Angioni, I. Furno, X. Garbet, C. Giroud, H. Leggate, P. Mantica, D. Mazon, J. Weiland, L. Zabeo, K.-D. Zastrow, and the JET-EFDA Contributors, Nucl. Fusion 45, L M. Greenwald, C. Angioni, J. W. Hughes, J. Terry, and H. Weisen, Nucl. Fusion 47, L H. Takenaga, K. Tanaka, K. Muraoka, H. Urano, N. Oyama, Y. Kamada, M. Yokoyama, H. Yamada, T. Tokuzawa, and I. Yamada, Nucl. Fusion 48, T. Fülöp, I. Pusztai, and P. Helander, Phys. Plasmas 15, C. Estrada-Mila, J. Candy, and R. E. Waltz, Phys. Plasmas 12, F. Romanelli and S. Briguglio, Phys. Fluids B 2, F. Romanelli, L. Chen, and S. Briguglio, Phys. Fluids B 3, S. C. Guo and F. Romanelli, Phys. Fluids B 5, M. A. Beer and G. W. Hammett, Phys. Plasmas 3, J. W. Connor, R. J. Hastie, and P. Helander, Plasma Phys. Controlled Fusion 48, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. Academic, New York, B. B. Kadomtsev and O. P. Pogutse, Rev. Plasma Phys. 5, Author complimentary copy. Redistribution subject to AIP license or copyright, see

108 Paper C I. Pusztai and P. J. Catto, Neoclassical plateau regime transport in a tokamak pedestal, Plasma Phys. Control. Fusion 52, (2010).

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126 Paper D I. Pusztai, S. Moradi, T. Fülöp, and N. Timchenko, Characteristics of microinstabilities in electron cyclotron and ohmic heated discharges, Phys. Plasmas. 18, (2011).

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128 PHYSICS OF PLASMAS 18, (2011) Characteristics of microinstabilities in electron cyclotron and ohmic heated discharges I. Pusztai, 1 S. Moradi, 1 T. Fülöp, 1 and N. Timchenko 2 1 Department of Applied Physics, Nuclear Engineering, Chalmers University of Technology and Euratom-VR Association, Göteborg, Sweden 2 Institute of Tokamak Physics, NRC Kurchatov Institute, , Kurchatov Sq. 1, Moscow, Russia (Received 28 March 2011; accepted 7 July 2011; published online 19 August 2011) Characteristics of microinstabilities in electron cyclotron (EC) and ohmic heated (OH) discharges in the T10 tokamak have been analyzed by linear electrostatic gyrokinetic simulations with GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] aiming to find insights into the effect of auxiliary heating on the transport. Trapped electron modes are found to be unstable in both OH and the EC heated scenarios. In the OH case the main drive is from the density gradient and in the EC case from the electron temperature gradient. The growth rates and particle fluxes exhibit qualitatively different scaling with the electron-to-ion temperature ratios in the two cases. This is mainly due to the fact that the dominant drives and the collisionalities are different. The inward flow velocity of impurities and the impurity diffusion coefficient decreases when applying EC heating, which leads to lower impurity peaking, consistently with experimental observations. VC 2011 American Institute of Physics. [doi: / ] I. INTRODUCTION Even if there is a wealth of experimental data and theoretical models relating to the effect of auxiliary heating on transport, there are still many open issues regarding the sign and magnitude of the transport and its parametric dependencies. One example of this is the experimental observation that the density profiles of electrons and impurities depends on the auxiliary heating. Results from many different devices have shown a flattening effect of electron cyclotron resonance heating (ECRH) on the electron density. 1 3 Furthermore, impurity accumulation also can be reduced by central ECRH. 4 6 However, in some parameter regions ECRH does not affect the electron or impurity density profiles, or even peaking of these profiles is observed. 7,8 The physical mechanism giving rise to the differences is not clearly identified, although it seems that collisionality plays a crucial role in determining the particle transport, 9 12 while the electronto-ion temperature ratio, 8 and the density and temperature scale lengths 7 are also important. In order to be able to make predictions confidently for future fusion devices, understanding of the underlying transport processes is necessary for a wide range of these parameters. In particular, to understand how and why the transport processes are different in the ohmic and electron cyclotron (EC) heated discharges it is important to analyze the turbulence characteristics and scalings with key parameters such as collisionality, electron-to-ion temperature ratio, and density and temperature scale lengths. The aim of the paper is to calculate the turbulence characteristics and the corresponding particle and energy fluxes for two similar experimental scenarios, one with ECRH and one with only ohmic heating. The steady-state impurity density gradient for trace impurities will also be calculated. As there is a consensus that the transport in tokamak core plasmas is mainly dominated by transport driven by drift wave instabilities, we focus on these instabilities and use quasilinear numerical simulations with the GYRO code 14 to calculate the turbulence characteristics. As shown in Refs. 15 and 16, the quasilinear electrostatic approximation retains much of the relevant physics and reproduce the results of nonlinear gyrokinetic simulations for a wide range of parameters. The experimental scenarios from T-10 are well-suited for the study we perform. One of the advantages is that there are measurements of turbulence characteristics on T-10 which can be used to compare with our theoretical calculations. The selected discharges have hot-electron plasma, relatively high density and collisionality; analyzing experimental scenarios from it gives insights into a range of parameters that have not been studied before. Our results show that the dominant instability is trapped electron (TE) mode in both the ohmic heated (OH) and the EC heated scenarios. As expected, the collision frequency plays an important role stabilizing the trapped electron mode driven turbulence in both cases. The EC heated case is more strongly suppressed for lower collisionalities. This is due to the drop of the electron temperature gradient drive which is stabilized by collisions. The growth rates and particle fluxes exhibit qualitatively different scalings with the electron-toion temperature ratios in the two cases. This is mainly due to the different collisionalities, but in the case of the electron particle flux also the difference in density gradients contributes. Sensitivity scalings for electron density and electron temperature gradients show that both of these drives are present in the investigated experimental scenarios. The inward flow velocity of impurities and the impurity diffusion coefficient decreases when applying EC heating, which leads to lower impurity peaking, consistently with experimental observations. The remainder of the paper is organized as follows. In Sec. II, the experimental scenario and theoretical modeling X/2011/18(8)/082506/8/$ , VC 2011 American Institute of Physics Author complimentary copy. 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129 Pusztai et al. Phys. Plasmas 18, (2011) are described. In Sec. III, the simulation results are presented and interpreted. We present the instability properties, the background ion and electron transport fluxes, and convective and diffusive transport of a trace impurity species. Finally, the results are summarized in Sec. IV. II. SCENARIO AND THEORETICAL MODELING A. Description of the T-10 discharges The T-10 plasma has circular cross-section with major radius R ¼ 1.5 m and minor radius a ¼ 0.3 m. The plasma current in the discharges was I p ¼ 200 ka and the toroidal magnetic field was B T ¼ 2.4 T. The effective charge in the comparatively high density discharges we studied was rather low Z eff ¼ 1.2. We study two typical experimental scenarios from T-10, one with 1 MW electron cyclotron heating, and one with only Ohmic heating. The plasma parameter profiles used in our study are given in Fig. 1. Carbon is an intrinsic impurity in all the discharges and here it is taken to be n C /n e ¼ 0.67% unless otherwise is stated. In discharges with on-axis EC heating, the electron density was found to decrease in the plasma center. There is sawteeth activity in the central part of the plasma in both discharges, and the inversion radius is around 7 cm. Therefore we will concentrate our studies outside r/a ¼ 0.4. B. Gyrokinetic modeling The linear gyrokinetic dynamics of all ion specii (deuterium, carbon, and a trace species with the concentration n Z /n e ¼ 10 5 ) and electrons has been simulated using the GYRO code. We consider only electrostatic fluctuations relevant for low b, and circular geometry. All the species are kinetic, and we include parallel compressibility and electronion collisions. We note that the ion-ion collisions were found to be unimportant even in the highest collisionality regions we studied. The carbon impurity is treated self-consistently. Unless otherwise specified, the following conventions and units are employed throughout this paper. Frequencies p and growth rates are normalized to c s /a, where c s ¼ ffiffiffiffiffiffiffiffiffiffiffiffi T e =m i is the ion sound speed, a is the plasma minor radius, and i is the main ion species. The fluxes are normalized to the flux surface average of k h q s je//t e j 2, where / is the amplitude of the fluctuating electrostatic potential, k h is the poloidal wave number, q s ¼ c s /x ci is the ion sound Larmor radius with x ci ¼ eb/m i the ion cyclotron frequency. The radial scale lengths are defined as L na ¼[@(ln n a )/@r] 1 and L Ta ¼[@(ln T a )/@r] 1, where a denotes the particle species. The magneto-hydrodynamic equilibrium, including the Shafranov shift of the circular flux surfaces, is calculated by the ASTRA code. 17 As the plasma rotation in the T-10 tokamak is weak it is neglected in our simulations. The linear GYRO simulations were carried out using fluxtube (periodic) boundary conditions, with a 128 point velocity space grid (8 energies, 8 pitch angles, and 2 signs of velocity), and the number of poloidal grid points along particle orbits is 14 for passing particles. The location of the highest energy grid point is at m i v 2 /(2T i ) ¼ 6. The electron-ion collisions were modeled by an energy dependent Lorentz collision operator. In the following section, we present the linear frequencies and growth rates from GYRO simulations to identify microinstabilities present in the experimental cases. Then for mid-radius and a representative wave number we perform parameter scalings to investigate the effect of collisions, temperature ratio, electron density, and ion temperature gradients on frequencies and linear particle and energy fluxes. Finally, the diffusion and particle flow and the zero flux density gradient of a trace impurity species is studied through impurity density gradient scalings. FIG. 1. (Color online) Safety factor (a), electron density (b), and ion and electron temperatures (c and d, respectively) for the two cases (OH solid, EC dashed). Author complimentary copy. Redistribution subject to AIP license or copyright, see

130 Characteristics of microinstabilities Phys. Plasmas 18, (2011) III. INSTABILITIES AND TRANSPORT A. Instability characteristics Figure 2 shows the growth rates and real frequencies of the instability as a function of k h q i at r/a ¼ 0.5 both in the collisionless case (c,d) and with collisions included (a,b). The linear simulations were performed for a range of k h q s - values using the Maxwell dispersion matrix eigenvalue solver method of GYRO to solve linear gyrokinetic equations. This method is capable of finding all the unstable roots of the Maxwell dispersion matrix, even the sub-dominant ones. The real part of the mode frequency is positive in all cases that suggests that the unstable modes are TE modes. This is confirmed by the fact that no unstable mode could be found if the non-adiabatic electron response was switched off. It is known that the non-adiabatic electron response can increase the ion temperature gradient mode growth rates, but the only unstable root is found to be a TE mode in both the OH and EC cases. This is in agreement with the experimental observations of the turbulent characteristics in high density cases similar to the ones studied here. 13 Trapped electron modes can be destabilized by both electron density and electron temperature gradients. 18 The normalized logarithmic density and temperature gradients at r/a ¼ 0.5 were a/l ne ¼ 1.47 and a/l Te ¼ 2.62 in the OH, and a/l ne ¼ 1.08 and a/l Te ¼ 3.96 in the EC case. One might expect that the higher growth rates in the EC case [dashed line in Fig. 2(a)] are due to the high value of a/l Te, but from Fig. 2(c) it becomes clear that the difference is mainly due to the effect of different collision frequencies, as without collisions the growth rates are rather similar. B. Effect of collisions It is interesting to note that the growth rate of the instabilities are reduced by the collisions as it can be seen on Fig. 2 comparing the upper (with collisions) and lower (without collisions) figures. This is due to the collisional de-trapping of trapped electrons and has been noted before in e.g., Ref. 20, where the growth rate of a dissipative trapped electron (DTE) mode in the long wavelength limit is found to be c e 3=2 x 2 e g e= ei, where ¼ r/r is the inverse aspect ratio, x *e ¼ k h T e /(ebl ne ) is the electron diamagnetic frequency, g e ¼ L ne /L Te, and ei is the electron-ion collision frequency. Our cases are quite similar to the DTE region, the parameter ( ei /)/jxj is typically much higher than one due to the modest temperatures and the high aspect ratio. On the other hand, the mentioned expression for the DTE growth rate 20 is obtained excluding magnetic drifts therefore we do not expect to find the same parametric dependence on collision frequency. The sensitivity of the growth rate and electron particle flux to the variation of the collision frequency in the OH and EC cases is illustrated on Figs. 3(a) and 3(b), respectively. The curves of the figure, as in all figures henceforth, correspond to mid-radius r/a ¼ 0.5 and k h q s ¼ 0.3. In previous trapped electron mode studies 18 it was found that the electron temperature drive of the TE modes is strongly suppressed as the collision frequency increases, while the density gradient drive can remain for higher collisionalities. From this perspective, although our experimental cases are not extreme examples for pure electron density or temperature driven TE modes, the OH case is more similar to the density gradient driven TE mode while the EC case having higher temperature gradient and lower density gradient is mainly driven by a/l Te. For lower collision frequencies, around the experimental value of ei in the EC case, the EC growth rate strongly decreases with increasing collisionality as the a/l Te drive is suppressed, but for higher collisionalities the mode is not completely stabilized due to the finite density gradient drive. The growth rate in the OH case or low collisionalities does not exhibit so strong dependence on ei,and that is what we expect in the density gradient driven TE case. FIG. 2. (Color online) Growth rates (a,c) and real frequencies (b,d) of the instability at r/a ¼ 0.5 (OH solid, EC dashed). Lower figures (c,d) are without collisions. Author complimentary copy. Redistribution subject to AIP license or copyright, see

131 Pusztai et al. Phys. Plasmas 18, (2011) FIG. 3. (Color online) Growth rate (a) and linear electron particle flux (b) of the instability obtained with linear GYRO calculations, as a function of electron-ion collision frequency. Solid curve: OH, dashed curve: EC. The markers correspond to the experimental value of the collision frequency (r/a ¼ 0.5, k h q s ¼ 0.3). The collisional stabilization of the modes also affects the electron particle fluxes [Fig. 3(b)], which exhibit qualitatively similar dependence on collision frequency as the growth rates for most of the plotted collisionality region. Regarding both the growth rates and particle fluxes, the experimental values are approximately the same in the two experimental scenarios, but this seems to be a coincidence considering the strong dependence of these quantities on collision frequency (and accordingly even stronger dependence on electron temperature). C. Temperature ratio effects FIG. 4. (Color online) T e /T i -scan of real frequencies (a) and growth rates (b) of the instabilities, the electron particle flux (c), and the ratio of ion and electron energy fluxes (d) for r/a ¼ 0.5 and k h q s ¼ 0.3. The OH case is shown by solid lines, the EC with dashed lines. It is reasonable to assume that one of the most important parameters that causes the differences between the OH and EC plasmas is the electron-to-ion temperature ratio. This parameter is indeed quite different for these cases; at r/a ¼ 0.5, T e /T i ¼ 1.01 in the OH, and T e /T i ¼ 1.95 in the EC case. However, the effect of T e /T i on the properties of the instability and the transport is qualitatively different in the two cases, as it can be seen in Fig. 4, where the real frequencies and growth rates of the instability are plotted together with the electron particle flux and the ratio of the ion and energy fluxes Q i /Q e as a function of T e /T i. [We note that these scalings are performed keeping the electron-ion collision frequency ð ei / Te 3=2 Þ fixed.] In the OH case the growth rate of the instability strongly decreases with this parameter almost on the whole temperature ratio region plotted, while the dependence is much weaker in the EC case, in the experimentally relevant regions. It is interesting to note that the slope of the EC growth rate curve is positive for the experimental value of the temperature ratio in the OH case. The slope of the C e (T e /T i ) curve is negative in the OH case while it is positive for the EC case. Thus, although the experimental electron particle flux values happen to be the same, the flux in the two cases exhibit qualitatively different behavior. The shape of the ion-to-electron energy flux ratio curves are similar in the two cases, the EC case being lower due to the higher electron energy flux corresponding to the stronger heating of electrons. In order to determine which parameter causes the qualitatively different T e /T i scalings between the OH and EC scenarios, we performed simulations where all parameters were identical to those of the OH case, except one, which we set to the corresponding value in the EC case. We expect that the TE mode growth rates are mainly affected by the density and electron temperature scale length and, as we saw in Sec. III B, the collisionality. The result of these simulations are shown in Fig. 5. The growth rate in the OH case (solid line, figure a) is mainly decreasing with T e /T i and this trend is even emphasized when we changed a/l Te (dotted) or a/l n (long dashed) to their value in the EC case. However, when the collision frequency was changed (dashed) the behavior of the growth rate curve became somewhat similar to that in the EC case (dash-dotted); the region for lower temperature ratios where the OH case showed increase in this parameter widened and the negative slope of the curve after the maximum growth rate is reduced. This effect of the collision frequency can be due to that for lower collisionalities, as it is in the EC case, the temperature gradient drive of the TE modes is more pronounced. One might expect that the difference in the particle fluxes have the same origin as for the differences in the growth rates; that would mean that the modified collisionality particle flux [dashed curve, Fig. 5(b)] should exhibit similar behavior to the EC flux (dash-dotted). This is partly true, as the positive slope region of the modified collisionality flux become somewhat wider and for higher values of T e /T i the negative slope of the curve decreased. However, changing the density gradient (long dashed curve) shifted the shape Author complimentary copy. Redistribution subject to AIP license or copyright, see

132 Characteristics of microinstabilities Phys. Plasmas 18, (2011) FIG. 5. (Color online) Electron-to-ion temperature ratio scalings of growth rates (a) and electron particle fluxes (b) for the original experimental cases (solid: OH, dash-dotted: EC) and in cases where all the parameters are taken from the OH case, except one, which is taken from the EC case. This parameter is chosen to be ei (dashed), a/l Te (dotted), and a/l n (long dashed). of the particle flux curve closest to the EC heated case. From this we can conclude that the qualitative behavior of the temperature ratio scaling of the growth rates is mainly affected by collisions, while for the electron particle flux the electron density gradient is also an important parameter from this aspect. D. Sensitivity to density and electron temperature gradients Figure 6 shows the real frequencies and growth rates of the instability, together with the electron particle flux and Q i /Q e as a function of the logarithmic density scale length a/l n. Recalling that the logarithmic density gradients were a/l ne ¼ 1.47 in the OH, and a/l ne ¼ 1.08 in the EC case, for r/a ¼ 0.5, we find that around these values the growth rate as a function of a/l n increases in both cases. This is not in contradiction with our previous statement that the EC case is more similar to a temperature gradient driven TE and the ohmic case is to a density gradient driven TE since both of them has some contributions from both drives. There is no qualitative difference in the C e (a/l n ) curves in the experimentally relevant region, but interestingly if we allow higher density gradients we find that above a/l n 2.2 the electron flux decreases with a/l n in the EC case, in contrast to the OH case which always drives higher particle flux for higher density gradient. This difference in the fluxes is related to the different behavior of the growth rates for higher density gradients; in the EC case c saturates, while it is steadily growing with a/l n in the OH case. The sensitivity of the energy flux ratio to the density gradients is more pronounced in the OH case than the EC case, but in both cases Q i /Q e (a/l n ) has a positive slope. Figure 7 shows the real frequencies and growth rates of the instability, together with the electron particle flux and Q i /Q e as a function of the logarithmic electron temperature scale length a/l Te. Keeping in mind that a/l Te ¼ 2.62 in the OH and a/l Te ¼ 3.96 in the EC case, we can see that the growth rate in the OH case are more strongly affected by a change in the electron temperature gradients than that in the EC case. This can seem somewhat counter-intuitive while we state that the EC case is mainly driven by the electron temperature gradient. However, we should consider that in both of these cases the TE mode is not exclusively driven by either of the gradients, but both of them with different weights; clearly, increasing the a/l Te in the OH case leads to that the case becomes more strongly driven by electron temperature gradients. Interestingly, although in the OH case the growth rate increases rapidly when the temperature gradient is increased and the Q i /Q e ratio is decreasing due to the higher electron FIG. 6. (Color online) a/l n -scan of real frequencies (a) and growth rates (b) of the instabilities, the electron particle flux (c), and the ratio of ion and electron energy fluxes (d) for r/a ¼ 0.5 and k h q s ¼ 0.3. The OH case is shown by solid lines, the EC with dashed lines. FIG. 7. (Color online) a/l Te -scan of real frequencies (a) and growth rates (b) of the instabilities, the electron particle flux (c), and the ratio of ion and electron energy fluxes (d) for r/a ¼ 0.5 and k h q s ¼ 0.3. The OH case is shown by solid lines, the EC with dashed lines. Author complimentary copy. Redistribution subject to AIP license or copyright, see

133 Pusztai et al. Phys. Plasmas 18, (2011) FIG. 8. (Color online) (a) Impurity convective flux V (arb. u., red curves) and diffusion coefficient D (arb. u., blue curves) for different impurity charge numbers (OH solid, EC dashed). (b) Impurity peaking factor. energy flux, the electron particle flux decreases with this parameter. In the EC case the electron particle flux is approximately constant over the plotted region, having a slight positive slope at the experimental value of a/l Te. E. Impurity peaking factor In similar experiments as the one studied here, a short argon gas-puff was applied in a stationary phase of both OH and EC heated discharges, and the evolution of the density of Ar þ 16 impurity (with density n z /n e 0.3%) was studied. 6 In discharges with on-axis EC heating, the argon density was found to decrease in the plasma center. The argon density reduction was proportional to the total input power. In this work we have studied the transport of a trace impurity with different charge in both EC and OH discharges. Our results indicate that the peaking factor becomes lower in the EC case, but it is still positive. In experimental work, the particle diffusivity is often separated into a diffusive part and a convective part, C z ¼D þ V zn z where D z is the diffusion coefficient and V z is the convective velocity. This separation of the flux into diffusion and convection can be done in the trace limit of impurity concentrations, because then the turbulence remains unaffected by the presence of the impurity, and the impurity flux varies linearly with the impurity density gradient. The particle flux of higher concentration minority ions, bulk ions, or electrons depends non-linearly on their density gradient as it can affect the growth rates of the mode. The amplitude of the perturbed quantities cannot be calculated in linear simulations, accordingly the convective velocity and diffusivity are plotted in arbitrary units in Fig. 8(a), still allowing the comparison of the two experimental cases. In Fig. 8(b) the steady state density gradient (or peaking factor) of impurities, a/l nz0 is shown. The peaking factor is calculated from the criterion C z (a/l nz0 ) ¼ 0 (assuming the non-turbulent particle fluxes and the impurity sources in the core negligible). All three quantities plotted in Fig. 8 exhibit a rather weak dependence on impurity charge. Furthermore for all three quantities, the ratio of their values for the two different experimental cases is almost independent of Z. The impurity peaking factor is approximately 2/3 times lower in the electron cyclotron heated case, consistently with the experimentally observed decrease of impurity content with EC heating. The inward pinch velocity is approximately 1/2 times lower in the EC heated case than in the OH case and the ratio of diffusion coefficients in the EC and the OH case is lower being around 1/3, which leads to the lower peaking factor in the EC case. The reduction of the peaking factor and in some cases even reversal of the impurity flux from inward to outward in the presence of ECRH has been noted before also in other experiments. 4,5 This was partly explained by the fact that the fluctuation of the parallel velocity of impurities along the field lines can generate an outward radial convection for TEmodes. 19 We can identify the contribution of this effect in our cases by switching of the parallel ion dynamics in the gyrokinetic simulations, see the corresponding plots of V, D and impurity peaking factor in Fig. 9. The difference between the two experimental cases is smaller without parallel ion dynamics which means that, indeed, the contribution of the parallel ion compressibility drives the system away from peaked impurity profiles. However, the value is positive for both cases, although in experiments the impurities FIG. 9. (Color online) Impurity transport without parallel ion compressibility. (a) Impurity convective flux V (arb. u., red curves) and diffusion coefficient D (arb. u., blue curves) for different impurity charge numbers (OH solid, EC dashed). (b) Impurity peaking factor. Author complimentary copy. Redistribution subject to AIP license or copyright, see

134 Characteristics of microinstabilities Phys. Plasmas 18, (2011) have a hollow profile corresponding to negative peaking factor. The reasons for the discrepancy might be due to that the effect of the Ware pinch is not considered, and we have only one wave number in a linear simulation instead of a whole range of interacting modes as in a nonlinear simulation. It should also be mentioned that gyrokinetic simulations perform usually better in terms of energy fluxes than for particle fluxes, 23,24 thus we do not expect perfect agreement between the experimental and simulated peaking factors. IV. CONCLUSIONS We compared the transport characteristics in electron cyclotron heated and purely ohmic plasmas on the T-10 tokamak using linear eigenvalue solver gyrokinetic simulations with the GYRO code. The aim was to obtain insights to the effect of electron cyclotron heating on the microinstabilities driving the turbulence, the corresponding particle and energy fluxes, and on the impurity particle transport. The only linearly unstable mode found in these experimental cases is a trapped electron mode. The frequency of collisional de-trapping is typically much higher than the mode frequency in these cases, accordingly the instabilities exhibit dissipative TE mode features; they are stabilized by collisions. However the modes are not completely stabilized by the collisions similarly to what was previously found in Ref. 18 for density gradient driven trapped electron modes. The higher linear growth rates found in the EC case are mainly due to the lower collision frequency in this high electron temperature plasma, and it is not an effect of the higher electron temperature gradient. The dependence of electron particle flux on T e /T i is qualitatively different in the two cases; in the OH case the electron particle flux decreases with this parameter, while it increases in the EC case. This behavior can be understood noting that the growth rate in the OH case decreases with increasing T e /T i, but in the EC case the dependence of the growth rate on this parameter is much weaker. The TE mode growth rate in the EC case strongly increases with increasing density gradient, while the growth rate in the OH case is almost independent on this parameter for the experimentally relevant region a/l ne In spite of the differences in the growth rate in this region, the electron particle flux shows qualitatively the same behavior in both cases. In TE mode dominated plasmas, as in our experimental scenarios, moving from pure ohmic to electron cyclotron heating leads to higher electron energy flux, since the collisional stabilization of the TE mode is less effective, and increasing the electron temperature gradient and electron-toion temperature ratio enhances the energy flux even further. On the other hand, the turbulent electron particle flux can remain approximately unchanged as the TE growth rate decreases with increasing electron-to-ion temperature ratio, balancing the opposite effect of the lower collisionality and the higher electron temperature gradient. It leads to the conclusion that the experimentally observed slight flattening of the electron density profile may have other reasons, e.g., the strength and period of sawteeth in the central region can be different and this can have implications on the density profile. The simulations indicate that the impurity convective flux is negative in both the EC and OH cases, but it is significantly lower in the EC case. Furthermore the impurity diffusion coefficient is lower in that case. As a consequence, the impurity peaking factor is lower in the EC case, however according to the simulations it does not change sign when electron cyclotron heating is applied. A sign change in the peaking factor is, therefore probably due to some additional physical mechanism, not accounted for in the linear gyrokinetic simulations. Recent work shows that impurity poloidal asymmetries may lead to a reduction or even sign change in the peaking factor. 21 Poloidal asymmetries may arise due to large pressure or temperature gradients if the plasma is sufficiently collisional, 22 and in this case it could be caused of the large temperature gradient due to EC heating. Finally, impurity accumulation is also affected by neoclassical processes, and the neoclassical impurity inward pinch is expected to be reduced in the presence of ECRH due to the flatter ion density profile. ACKNOWLEDGMENTS The authors gratefully acknowledge helpful conversations with V. Krupin and V. A. Vershkov, and would like to thank J. Candy for providing the GYRO code. The authors acknowledge the work of the T-10 experimentalists, who provided the information about plasma parameters. This work was funded by the European Communities under Association Contract between EURATOM and Vetenskapsrådet. The views and opinions expressed herein do not necessarily reflect those of the European Commission. One of the authors, S.M., acknowledges the support from the Wenner- Gren Foundation. 1 P. Gohil, L. R. Baylor, K. H. Burrell, T. A. Casper, E. J. Doyle, C. M. Greenfield, T. C. Jernigan, J. E. Kinsey, C. J. Lasnier, R. A. Moyer, M. Murakami, T. L. Rhodes, D. L. Rudakov, G. M. Staebler, G. Wang, J. G. Watkins, W. P. West, and L. Zeng, Plasma Phys. Controlled Fusion 45, 601 (2003). 2 H. Takenaga, S. Higashijima, N. Oyama, L. G. Bruskin, Y. Koide, S. Ide, H. Shirai, Y. Sakamoto, T. Suzuki, K. W. Hill, G. Rewoldt, G. J. Kramer, R. Nazikian, T. Takizuka, T. Fujita, A. Sakasai, Y. Kamada, H. Kubo, and the JT-60 Team, Nucl. Fusion 43, 1235 (2003). 3 A. Zabolotsky, H. Weisen, and TCV Team, Plasma Phys. Controlled Fusion 48, 369 (2006). 4 R. Dux, R. Neu, A. G. Peeters, G. Pereverzev, A. Muck, F. Ryter, J. Stober, and ASDEX Upgrade Team, Plasma Phys. Controlled Fusion 45, 1815 (2003). 5 E. Scavino, J. Bakos, H. Weisen, and TCV Team, Plasma Phys. Controlled Fusion 46, 857 (2004). 6 N. Timchenko, V. Vershkov, V. Karakcheev, D. Shelukhin, A. Dnestrovskij, V. Krupin, A. Gorshkov, D. Ryzhakov, and I. Belbas, The effect of EC heating on impurity transport in T-10, in Proceedings of the 34th EPS Conference on Plasma Physics, Warsaw, C. Angioni, A. G. Peeters, X. Garbet, A. Manini, F. 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135 Pusztai et al. Phys. Plasmas 18, (2011) 12 H. Takenaga, K. Tanaka, K. Muraoka, H. Urano, N. Oyama, Y. Kamada, M. Yokoyama, H. Yamada, T. Tokuzawa, and I. Yamada, Nucl. Fusion 48, (2008). 13 N. Timchenko, V. Vershkov, V. Karakcheev, S. Krasnjanskii, V. Krupin, D. Shelukhin, V. Merejkin, D. Sarichev, A. Gorshkov, and I. Belbas, Experimental study of particles and heat transport in T-10 Ohmic plasmas, in Proceedings of the 36th EPS Conference on Plasma Physics, Sofia, J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003). See fusion.gat.com/theory/gyro. 15 T. Dannert and F. Jenko, Phys. Plasmas 12, (2008). 16 A. Casati, C. Bourdelle, X. Garbet, F. Imbeaux, J. Candy, F. Clairet, G. Dif-Pradalier, G. Falchetto, T. Gerbaud, V. Grandgirard, Ö. D. Gürcan, P. Hennequin, J. Kinsey, M. Ottaviani, R. Sabot, Y. Sarazin, L. Vermare, and R. E. Waltz, Nucl. Fusion 49, (2009). 17 G. V. Pereverzev and P. N. Yushmanov, ASTRA Automated System for TRansport Analysis, Max-Planck-Institut Für Plasmaphysik, IPP- Report, IPP 5/98, C. Angioni, A. G. Peeters, F. Jenko, and T. Dannert, Phys. Plasmas 12, (2005). 19 C. Angioni and A. G. Peeters, Phys. Rev. Lett. 96, (2006). 20 J. W. Connor, R. J. Hastie, and P. Helander, Plasma Phys. Controlled Fusion 48, 885 (2006). 21 T. Fülöp and S. Moradi, Phys. Plasmas 18, (2011). 22 T. Fülöp and P. Helander, Phys. Plasmas 8, 3305 (2001). 23 C. Angioni, E. Fable, M. Greenwald, M. Maslov, A. G. Peeters, H. Takenaga, and H. Weisen, Plasma Phys. Controlled Fusion 51, (2009). 24 R. E. Waltz, A. Casati, and G. M. Staebler, Phys. Plasmas 16, (2009). Author complimentary copy. Redistribution subject to AIP license or copyright, see

136 Paper E I. Pusztai, J. Candy, and P. Gohil, Isotope mass and charge effects in tokamak plasmas, accepted for publication in Phys. Plasmas.

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161 Paper F I. Pusztai, G. Pokol, D. Dunai, D. Réfy, G. Pór, G. Anda, S. Zoletnik and J. Schweinzer, Deconvolution-based correction of alkali beam emission spectroscopy density profile measurements, Rev. Sci. Instrum. 80, (2009).

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163 REVIEW OF SCIENTIFIC INSTRUMENTS 80, Deconvolution-based correction of alkali beam emission spectroscopy density profile measurements I. Pusztai, 1 G. Pokol, 2 D. Dunai, 3 D. Réfy, 2 G. Pór, 2 G. Anda, 3 S. Zoletnik, 3 and J. Schweinzer 4 1 Department of Radio and Space Science, Chalmers University of Technology, EURATOM-VR Association, SE Göteborg, Sweden 2 Department of Nuclear Techniques, Budapest University of Technology and Economics, Association EURATOM, Műegyetem rkp. 9., H-1111 Budapest, Hungary 3 KFKI-RMKI, Association EURATOM, Pf. 49, H-1525 Budapest, Hungary 4 Max-Planck-Institut für Plasmaphysik, Association EURATOM, D Garching, Germany Received 12 February 2009; accepted 26 July 2009; published online 20 August 2009 A deconvolution-based correction method of the beam emission spectroscopy BES density profile measurement is demonstrated by its application to simulated measurements of the COMPASS and TEXTOR tokamaks. If the line of sight is far from tangential to the flux surfaces, and the beam width is comparable to the scale length on which the light profile varies, the observation may cause an undesired smoothing of the light profile, resulting in a non-negligible underestimation of the calculated density profile. This effect can be reduced significantly by the emission reconstruction method, which gives an estimate of the emissivity along the beam axis from the measured light profile, taking the finite beam width and the properties of the measurement into account in terms of the transfer function of the observation. Characteristics and magnitude of the mentioned systematic error and its reduction by the introduced method are studied by means of the comprehensive alkali BES simulation code RENATE American Institute of Physics. DOI: / I. INTRODUCTION Lithium beam emission spectroscopy Li-BES is an active diagnostic tool, typically probing the outer regions of fusion plasmas by observing the characteristic emission of a kev atomic beam injected into the plasma. 1,2 Li-BES measurements are routinely performed on several fusion devices. 3 Application of sodium for BES purposes is recently considered, 4,5 thus we refer to the method as alkali BES hereafter. The evolution of the populations of different atomic states depends on the distribution of plasma parameters along the beam line. This means that from the emitted intensity distribution at a characteristic frequency i.e., the light profile information can be obtained on the distribution of electron density 6 and its fluctuation. 2 Advantages of the alkali BES diagnostic include being practically nonintrusive because of the low density of the beam and being approximately a point measurement. The technique is exceedingly suitable for scrape-off layer SOL and pedestal density profile measurements with good temporal and spatial resolution 50 s, 5 mm, 7 thus contributes to the understanding of transitions between different confinement modes and to the validation of models of edge transport barrier formation. 8 Statistical behavior of density fluctuations with approximately microseconds time scale in the edge and SOL e.g., radial wave number spectra, correlation length and time can also be investigated in the radial direction by single beam fluctuation measurements. 2 Moreover, fluctuation measurements can be extended to two dimensions by electrostatically deflecting the beam in the poloidal plane in the fluctuation s time scale in order to obtain a complex picture of the cross field turbulent transport. 9 We have to point out that the high temporal resolution fluctuation measurements need always to be combined with the significantly slower profile measurement. The collisional-radiative model is considered in the standard description of beam evolution. 10 Based on this model, reliable numerical methods were developed to determine the density profiles along the beam line, given the observed light profile, 7,11,12 such as the Li-BES density reconstruction code ABSOLUT, which is used in the present study. 6 All these methods are based on the assumption that the measured light profile is equivalent to the emissivity of the beam. This case corresponds to the ideal measurement geometry considering the beam to be one-dimensional 1D and neglecting its finite width. Our goal is to provide a tool to quantitatively measure the systematic error caused by this simplified treatment. Partially for this purpose, as well as to support the design and interpretation of BES measurements, the RENATE alkali BES simulation code has been developed. For the purpose of this study, alkali BES setup and plasma parameters are chosen from the recently upgraded TEXTOR Li-BES diagnostic, 13 and the alkali BES diagnostic planned for the newly restarted COMPASS. 14 We investigate the character and magnitude of the systematic error in density profile reconstruction, and conclude that it can be significant in certain, experimentally relevant cases, which we support with a general estimation of the maximal error in the calculated electron density. The simulation of the phenomenon also enabled the design of a method for the correction of the measured light profile re /2009/808/083502/8/$ , American Institute of Physics Downloaded 27 Aug 2009 to Redistribution subject to AIP license or copyright; see

164 Pusztai et al. Rev. Sci. Instrum. 80, ducing the effect of the finite beam width, and thus allowing the use of the 1D density reconstruction methods. The structure of the paper is as follows: In Sec. II the alkali BES measurement simulation, RENATE is introduced. The issues of the observation of a finite width beam are investigated in Sec. III. The emission reconstruction correction method is discussed in Sec. IV, and demonstrated through realistic simulated measurements in Sec. V. A general estimate of the error due to finite beam width is given in Sec. VI, and finally, the results are summarized in Sec. VII. II. RENATE ALKALI BES MEASUREMENT SIMULATION For the purpose of supporting the design of alkali BES density profile and fluctuation measurements, an Interactive Data Language IDL simulation code, RENATE, has been developed which also assists the interpretation and correction of measured data. In order to take the finite width of the beam into account, the beam evolution is calculated separately in slices of the beam considering a realistic current distribution. The integration of emitted light along the lines of sight is modeled together with other essential features of the observation and the detector system. The atomic physics processes of the beam are modeled by the collisional-radiative model. 10 The rate equations describing the evolution of atomic occupations can be written in a quite compact form dn i dx = n e xã ij x + b ij n j x i, j =1,...,m, 1 j where n e is the electron density, n i and n j are the populations of the i th and j th atomic states, respectively, and x is the coordinate along the beam. The atomic transition and electron loss processes due to electron e, proton p, and impurity I collisions are described by the reduced rate coefficient matrix ã ij. 4,15 17 Taking the effect of the impurities into account through one representative impurity characterized by charge qx and producing an effective ion charge Z eff x, the matrix is written as ã ij =a e ij +1 qfa p ij + fa I ij, where f =Z eff 1/qq 1. The spontaneous atomic transitions are described by the b ij matrix. In the simulation, the numbers of registered atomic levels are m=9 for lithium and m=7 for sodium. Note that ã ij depends on x not only through q and Z eff but due to the temperature dependence of the rate coefficients. The ion and impurity temperatures are chosen to be equal to the electron temperature, causing only a negligibly small error, due to the flat temperature dependence of a p ij and a I ij. The photon emission density of a beam per unit time and length is proportional to n IA /v B, where the observed spectroscopic line corresponds to the transition 2p 2s for Li, 3p 3s for Na, v B is the beam velocity, A is the corresponding Einstein coefficient, and I is the beam current. The n population is calculated by the solution of the direct problem, integrating Eq. 1 stepwise from the point where the beam enters the plasma x=0, with the initial condition n i 0= i1, where 1 is the index of the ground state. Since the RENATE code was originally developed with the purpose of design of BES measurements, it solves the direct problem calculating the beam evolution and the emission distribution for a given measurement configuration and set of plasma parameters. Therefore, the most important component of the simulation is its atomic physics kernel, which calculates the rate coefficients from parametrically given cross sections of the collisional processes and solves the rate of Eq. 1 by a fourth order Runge Kutta method. The spontaneous atomic transition probabilities are taken from the National Institute of Standards and Technology NIST atomic spectra database, 18 and the cross section data found in Refs for lithium and Ref. 4 for sodium are used. The collisional j i de-excitation rate coefficients are derived from the corresponding i j excitation rate coefficients using the principle of detailed balance, while impurity collision rate coefficients are calculated from the proton collisional cross sections using the scaling relations given in Refs. 4 and 15. The proton impact target electron loss processes are considered instead of treating the ionization and charge exchange channels separately, which is the main difference of the atomic physics kernel from the ABSOLUT Ref. 6 inverse problem solver regarding the atomic physics. ABSOLUT has a corresponding direct solver code called SIMULA, which we used for the validation of RENATE. The found relative difference between the rate coefficients calculated by the different programs is O10 4, and accordingly the maximum relative difference between the calculated evolution of atomic populations is the same order of magnitude. The ABSOLUT code in turn has been critically tested against both Li Ref. 6 and Na Ref. 5 measurements. In this manner, RENATE is indirectly validated to measurements; the direct validation is under way at the TEXTOR tokamak. The calculation scheme of the simulation is as follows. First, the beam is divided into slices which are perpendicular to the poloidal plane while the velocity of the beam atoms is tangential to them. The emissivity profile is calculated along each slice, given the magnetic geometry R,Z, together with the distribution of the relevant plasma parameters, n e, T e, q, and Z eff as a function of a flux coordinate. The plasma parameters are assumed to be equally distributed on a flux surface. Then the calculation of the geometric efficiency i.e., the effect of that the collecting optical element covers different solid angles seen from different points of the beam is performed for the points of each slice, and the efficiency of the observation system is taken into account in order to determine the number of photons per unit time detected by each detector segment. Contributions of the different beam slices and points to the detected signal are summed up. We restrict our studies to measurement geometries where the beam axis is in the poloidal plane and the observation point is also located in the same toroidal position, which is typical for diagnostic neutral beams. In this case, we can project the three-dimensional beam into the poloidal plane of the beam axis, and the observed volumes reduce to observed areas. III. OBSERVATION OF A FINITE WIDTH BEAM The observed signal from a diagnostic beam is equal to the integral, along the line of sight, of the emissivity Downloaded 27 Aug 2009 to Redistribution subject to AIP license or copyright; see

165 Pusztai et al. Rev. Sci. Instrum. 80, a FIG. 1. Color online Construction of the transfer function of the observation. x is the coordinate along the beam axis, which is one-to-one mapped to x through the lines of sight crossing the axis. The image Sx of a light source being on the flux surface poked by the axis at x gives Tx,x=x. weighted by the geometric efficiency. Since the beam has a finite width, a line of sight goes through parts of the beam being in different stages of beam evolution, thus the measurement cannot be perfectly local. Inverting this effect, the emission reconstruction method gives an estimate of the emissivity on the beam axis from a measured light profile. We denote the coordinate measured along the beam axis by x, and index each segment of the detector array by x, marking the position where the middle of the observed volume of the detector segment intersects the beam axis see Fig. 1. For the sake of simplicity of the formalism, x is also considered to be a continuous independent variable. Assuming that the plasma parameters are flux functions, it can be concluded from our simulations that the evolution of atomic populations also follows the flux surfaces, except from extreme cases of wide beams injected almost tangentially to the flux surfaces. This enables us to extend the emissivity along the beam axis Ix into two dimensions by mapping along the flux surfaces indexed by x marking their intersection with the beam axis and weighting with the beam current distribution. Thus, we can express the measured light profile Sx as Sx = Tx,xIxdx, where the kernel function Tx,x is called the transfer function of the observation. Obviously, the goal is to determine Ix from a measured Sx. Equation 2 suggests the way to calculate the transfer function Tx,x, since the choice of the emissivity Ix =x x gives Sx=Tx,x, where x scans all possible values of x. For a given measurement configuration, the Tx,x transfer function of observation can indeed be calculated by simulating the observation of virtual light sources on the x flux surfaces for a whole range of x values, as it is illustrated in Fig. 1. In Fig. 2, two transfer functions are contour plotted, illustrating the features of the deviation from an ideal measurement that would give a x x-like transfer function fully centered upon the diagonal. Figure 2a corresponds to an unfavorable setup, when the lines of sight are quite far from tangential to the flux surfaces, in contrast to Fig. 2b. A horizontal cut of the former transfer function is a wide Gaussian-type curve, showing that a detector segment in a given x position collects the information from a broad range 2 b FIG. 2. Simulated transfer functions. The observation angle is a high or b small. of spatial coordinate x. The closer the lines of sight to tangential are to the flux surfaces at the beam position, the more local the measurement is. The effect caused by the broadening of the transfer function, due to finite thickness of the beam, on the measured light profile and the corresponding density profile is illustrated for a quite unfavorable but still realistic case. The angle between the lines of sight and the flux surfaces, which we call observation angle, is approximately 45 on average at the beam position. The observation system is located 0.45 m far from the observed region. A Gaussian current distribution beam with full width at half maximum of 1.2 cm is injected into a high density plasma with pedestal. For this case the transfer function is similar to Fig. 2a, and the corresponding light profiles are presented in Figs. 3a and 3b. The emissivity distribution of an infinitesimally fine beam Ix would give the best measurement of the electron density on the beam axis, up to the accuracy of the density profile reconstruction method. This strictly local measurement is referred as ideal solid line. In reality we measure a light profile measured, dotted line affected by the finite beam width, which is smoother compared to the ideal. Before the density calculation, the measured profile is corrected to the spatially slowly varying geometrical efficiency factor giving the profile labeled as calibrated dashed line. Note that the density reconstruction does not require the absolute value of the emissivity, only the shape of the light profile. The relative differences from the ideal profile with respect to the maximum intensity are plotted in Fig. 3b. Note that while the relative difference between the ideal and the Downloaded 27 Aug 2009 to Redistribution subject to AIP license or copyright; see