Study of magnetic island propagation and evolution in tokamak plasmas

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Universit´

a degli Studi di Pisa

Corso di Dottorato di Ricerca in Fisica

XXIX Ciclo (2013-2016)

PhD Thesis

STUDY OF MAGNETIC ISLAND

PROPAGATION AND EVOLUTION IN

TOKAMAK PLASMAS

Candidate:

Andrea Casolari

Supervisors:

Dr. Paolo Buratti

Prof. Francesco Pegoraro

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Aknowledgements

I would like to thank the FUSPHY division of FSN department (C.R. ENEA in Frascati) for the hospitality during the two years of my thesis work that I spent there.

I am thankful to my supervisor Dr. Paolo Buratti for introducing me to the subject of magnetic islands and for sharing with me his deep knowledge on this complex and fascinating issue in plasma confinement. I also want to thank him for getting me involved in the experimental work performed in the Frascati tokamak FTU.

I have to thank Prof. Francesco Pegoraro for teaching me the importance of having a good physical intuition and for helping me understand the most difficult aspects of the theories I had to study to progress in my work.

I have to thank Dr. Daniela Grasso from ISC-CNR and Politecnico di Torino for lending me the code developed by her and colleagues to perform the simulations we used in this thesis. I also want to thank her for the valuable help in the acquisition and the interpretation of the numerical results.

I want to thank my friend Matteo Valerio Falessi from Universit´a degli Studi Roma Tre for the precious input I received from our valuable discussions on plasma physics related issues.

Finally I would like to express my gratitude to my parents for supporting me through this ordeal, as well as in every other enterprise I have tried to carry out.

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Nomenclature

ω∗e Electron diamagnetic frequency ω∗e= −Te/(eBLn).

ω∗i Ion diamagnetic frequency ω∗i= Ti/(eBLn).

W Island width W = 2pLsψ/B.

Ω Ion cyclotron frequency Ω = eB/mi.

ρs Ion-acoustic radius ρ2s = Te/(miΩ2).

ρi Ion Larmor radius ρ2i = Ti/(miΩ2).

νi Ion collision frequency νi = ne4log Λ/(m2ivthi3 ).

ωb Bounce frequency ωb = [H dθ/(vkb · ∇θ)]−1.

q Safety factor q = hB · ∇ϕi / hB · ∇θi. Inverse aspect ratio = r/R0.

β Ratio between plasma pressure and magnetic pressure β = P/(B2_/2µ 0).

ν∗ Collisionality parameter ν∗ = νi/(ωb).

τR Resistive time τR = L2/η.

τA Alfv´en time τA= (min0)1/2/(kB).

∆0 Linear stability parameter ∆0 = (ψ_up0 (0+) − ψ0_dw(0−))/ψ(0). S Lundquist number S = τR/τA.

cθ Intrinsic poloidal rotation coefficient cθ = 1.17ηi, ηi = Ln/LT.

νθ Poloidal flow damping coefficient νθ = ftνi(qs/s)2.

c⊥ Intrinsic toroidal rotation coefficient c⊥= 2.37ηi.

ν⊥ Perpendicular flow damping coefficient ν⊥ = ftνi(qs/s)2(nφ/ν∗)2(W/R0)2.

τ Ratio between the ion and the electron temperature τ = Ti/Te.

α2 _{Ion-acoustic effects parameter α}2 _{= W}2_L2

n/(ρ2sL2s).

ρ2 _{Small-Larmor-radius parameter ρ}2 _{= ρ}2 i/W2.

Ln Density gradient length Ln= |d log n/dr|−1.

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Chapter 1 Introduction

Magnetic confinement is the attempt to obtain a self-sustained fusion reaction inside a plasma by using magnetic fields. A plasma can be defined as a collection of charged particles, globally neutral and whose dynamics is dominated by collective phenomena instead of single-particle interactions. In a non-magnetized plasma, particles move randomly in all directions under their thermal motion; in a magnetized plasma, instead, charged particles follow the magnetic field lines but for their small Larmor radius. Also, collisions and drift motions cause the particles to deviate from the field lines. In a cylindrical device, with a homogeneous magnetic field directed along the symmetry axis, particles will move mostly along the axial direction. To prevent the particles from escaping from the ends of the device, the cylinder has to be bent into a torus, so that the particles are constrained to travel along the toroidal direction, ideally never reaching the walls of the device. In spite of the advantage of having closed magnetic field lines, the toroidal geometry introduces drift motions and trapped particle orbits, which leads to the so-called neoclassical effects, because of the inhomogeneity of the magnetic field. The most performing device currently used in the quest for fusion energy is the tokamak, that is a toroidal chamber where plasma is contained, while a toroidal magnetic field is imposed by external coils. An external transformer induces in the plasma a toroidal current, which in turn generates a poloidal magnetic field, adding to the toroidal one. The helical magnetic field, resulting from the sum of toroidal and poloidal magnetic fields, provides confinement and stability. A schematic representation of the magnetic field in a tokamak is provided in Fig.1.1.

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At equilibrium, magnetic field lines lie on surfaces forming a family of nested tori, named magnetic surfaces. For the reasons mentioned above, particles can move freely along the mag-netic surfaces, but their motion is strongly limited in the perpendicular direction. This sig-nificantly reduces the radial transport of heat and particles, thus providing a good energy and particle confinement, which is essential for reaching a high temperature in the core of the plasma. This structure of nested magnetic flux surfaces can be affected by instabilities. One of the most important ones is the so-called tearing mode, which is an instability ”tearing” and re-connecting magnetic field lines. Magnetic reconnection locally breaks the topology of magnetic surfaces leading to a more energetically-favorable configuration. Magnetic islands result from the nonlinear evolution of tearing modes and represent a serious obstacle for obtaining nuclear fusion in magnetic confinement devices because they locally break the magnetic surfaces, thus allowing particles to escape more easily from the centre to the walls of the tokamak. The uncon-trolled growth of magnetic islands can also lead to major disruptions, causing serious damage to the device. Magnetic islands are an important issue in the physics of plasma confinement and they are the subject of this thesis work.

1.1 Background and motivations

Many efforts have been done in the past decades to develop a theory of magnetic island dynamics in tokamaks. The interest in this kind of studies is to understand the conditions for the onset of the islands in the tokamak experiments and to control them to prevent their growth to large amplitudes and the consequent negative effects on confinement. Also a great deal of interest is addressed to the interpretation of the observations of steady-state rotation of the islands and on the rapid variations in their dynamics. Magnetic islands arise from the nonlinear evolution of tearing modes [34]. In the presence of an equilibrium density and temperature gradient, the tearing mode acquires a propagation frequency and the instability is said a drift-tearing mode [2]. According to the linear drift-tearing dispersion relation, the propagation frequency of the instability should be close to the electron diamagnetic frequency, ω − ωE ≈ ω∗e [20],[2], where

the frequency is related to the velocity through the wave vector k, ω = k·v. The tearing mode is an instability characterized by a long wavelength, which corresponds to a small wavevector. ωE

is the E ∧ B-drift frequency, due to the equilibrium electric field. In fact, the plasma as a whole rotates with the E ∧ B velocity, so that this contribution must be subtracted from the island rotation velocity (Doppler shift). Experimental observations of magnetic islands in tokamaks, under specific conditions, show a rotation frequency closer to the ion diamagnetic frequency, ω − ωE ≈ ω∗i [79],[13],[12]. The measurement of the E ∧ B velocity in [13],[12] was made

possible by the availability of different diagnostic techniques, namely Motional Stark Effect (MSE), high-resolution Thomson scattering (HRTS) and charge-exchange (CX) recombination spectroscopy. Because of this disagreement between the predictions of the linear theory and the experimental observations, we have compared the most credited theoretical models describing magnetic island dynamics, trying to understand if they could be improved in order to match the experimental observations, in particular those performed on the Joint European Torus (JET) [13],[12]. According to recently developed models, in the presence of significant electron temperature gradients, the introduction of the so-called ”mode inductivity” [21] in the Ohm’s law permits the existence of modes propagating with the ion diamagnetic frequency. This

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effect arises naturally in the linear regime, but the experimental observations of the island rotation concern nonlinear islands, thus a direct check of the validity of this model is not currently possible. Another widely accepted interpretation of the observed rotation velocity is that, when the island width becomes larger than the ion-acoustic radius, the ion fluid cannot cross the island separatrix, which is the ideal magnetic surface separating the intact and the reconnected field lines, thus the island is forced to propagate with the velocity of the ion flow [27]. This explanation works for islands which are large enough, but it cannot account for the transition from one diamagnetic velocity to the other. Nonlinear island dynamics is still not fully understood, and the processes that determine the island rotation velocity are under investigation.

Attempts to study the stationary rotation of magnetic islands have been made by Fitzpatrick & Waelbroeck in a series of papers on the subject [27],[31],[28],[29] by solving an improved version of the four-field model, previously deduced by Hazeltine, Kotschenreuther and Morrison [43], which is a reduction of the two-fluid plasma description. Among other things, the authors studied the effect of the island width on the evolution of the quantities characterizing the plasma, such as density and electrostatic potential, near a magnetic island, for the role these quantities play on the island stability and on it’s velocity. They also studied the effect of the neoclassical flow damping on the island rotation by including in the equations a simplified expression for the flow damping caused by neoclassical effects, as it was obtained by several authors in previous works [45],[47],[70],[68]. The result of their studies is that both the island width and the neoclassical effects influence the island rotation. In particular, the critical parameters which determine the island dynamics are the ratio between the island width W and the ion-acoustic radius ρson one side, and the ratio between the collision frequency νi and the bounce frequency

ωb on the other side. The first dimensionless ratio determines if the island is in the sonic or

hypersonic regime, which is related to the relative role of ion-acoustic waves on the flattening of the density profile inside the separatrix. The second dimensionless parameter determines if the plasma is in the weak or in the intermediate damping regime, which is related to the relative strength of the neoclassical effects. The simultaneous presence of both the effects in a tokamak plasma makes it particularly difficult to determine the islands rotation velocity.

Also, the observation of a new phenomenon involving magnetic islands in Frascati Tokamak Upgrade (FTU) has attracted our interest. This phenomenon was given the name ”limit cycles” [61] because of the figure drawn by the trajectory of a point in the plane having on the axes the amplitude and the frequency of magnetic signals produced by rotating islands. In this regime of propagation, which occurs only in the high-density discharges with high density and temperature gradients, the magnetic islands show a rapid variation of their amplitude and rotation frequency, which gives to the phase portrait the characteristic shape of a closed curve. Unlike in the measurements performed in JET, the island rotation frequency in FTU refers to the laboratory frame of reference because the diagnostic for the measurement of the radial electric field is not available. The rapid dynamics shown by the islands in this regime raises questions on the validity of the widely accepted model formulated by Rutherford [64] on the nonlinear dynamics of magnetic islands, prescribing a characteristic growth time of the order of the much longer resistive diffusion time.

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1.2 Purpose of this work

The aim of this thesis work is to investigate the magnetic island dynamics in tokamaks, in par-ticular as regards island rotation. The attempts to study the island rotation by Fitzpatrick & Waelbroeck rested on the inclusion of the neoclassical effects in their equations by using simpli-fied expressions for the neoclassical terms, together with the possibility to keep the island-size effects by using an appropriate normalization for the fields. Although their work shows results consistent with the experimental observations, their results come from a system of fluid equa-tions which did not include the Finite Larmor Radius (FLR) correcequa-tions from the start. In this work we attempt to improve their results by starting from a set of gyrofluid equations, which result from taking the moments of the gyrokinetic equation [10],[24], and then reducing them to a four-field model analogous to that used by Fitzpatrick & Waelbroeck. The neoclassical effects are included in the model by using the same simplified expressions, with an important difference. To be consistent with the gyrofluid equations, we compute the lowest order FLR corrections to the poloidal flow damping by solving the gyrokinetic equation in an appropriate limit and then computing the poloidal flow damping by following the same approach adopted in the book ”Collisional transport of impurities in plasmas” by Helander & Sigmar [45]. The equations thus obtained have been solved by adopting a series of perturbative expansions in-troduced by Fitzpatrick & Waelbroeck in their works and based on the multiple-scale approach [49]. The final equations have been solved, in two different regimes of collisionality, together with the torque balance condition, imposing that the total electromagnetic force acting on the freely-rotating islands is zero. The solution of this system of equations provides the field profiles and the self consistent phase velocity of the islands. Attempts to study both analytically and numerically the FLR effects on magnetic island evolution have been done [81],[71], finding again several results which were expected for small islands, which cannot be studied accurately with a purely fluid model. Still, in these works the focus of the authors was mainly on the analysis of the island dynamics and related phenomena, such as the emission of drift waves and the flattening of the density profile, given the island phase velocity, which was just a parameter of their models. The approach we choose, which is the same used by Fitzpatrick & Waelbroeck, is significantly different because our purpose is to deduce the island rotation frequency consis-tently with the field profiles in a stationary regime.

The system of four-field gyrofluid equations we have deduced generalizes the four-field model adopted by Fitzpatrick & Waelbroeck in their works and, in particular, the inclusion of the lowest-order FLR corrections to the poloidal flow damping leads to an additional term in the intermediate-damping case (which is relevant for realistic tokamak applications). The numeri-cal resolution of the final system of equations, together with the torque-balance condition, in principle, provides the island rotation velocity for different values of the plasma parameters, within the limits of validity of the model.

Furthermore, we have investigated the above mentioned phenomenon of ”limit cycles”, recently observed in FTU. By numerically integrating a modified version of the four-field system of equations by using a finite-difference slab code developed by Daniela Grasso et al. from ISC-CNR and Politecnico di Torino [39], we have discovered that an island deformation occurs in the presence of a strong diamagnetic velocity. The deformation becomes significant in the non-linear regime, when the island width becomes large enough to flatten the density profile inside the separatrix, so that the diamagnetic velocity of the plasma is zero inside the separatrix and

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becomes different from zero just outside, causing a velocity shear which causes the island de-formation. The diamagnetic velocity shear in the nonlinear regime also causes a reversal in the island growth, which is compatible with the cycle dynamics [16]. To have a more sound evidence of the validity of our hypothesis, we should be able to observe the same periodic behavior in the frequency. We were unable to confirm this fact, although the results we have collected so far are promising. The numerical analysis of the rotation frequency is ongoing.

1.3 Outline of the thesis

In Chapter 2 we introduce the toroidal geometry and we review the basic concepts of neoclassical theory, originally developed to deal with transport phenomena in tokamaks. Then we show the methods which have been used to solve the drift-kinetic equation, with the purpose of deriving analytical expressions for the poloidal flow damping. In Chapter 3 we review the theory of tearing modes and magnetic islands, both in the linear and the nonlinear regime. We also show our attempt to include the effect of the poloidal flow damping on the linear growth rate. In Chapter 4 we review the gyrokinetic and gyrofluid theories, which have long been used to study turbulence in plasmas. In Chapter 5 we show our analytical calculations, which lead to a system of gyrofluid equations including the neoclassical effects. In Chapter 6 we present the observations of the ”limit cycle” activity in FTU and the numerical results obtained by integrating a modified version of the four-field system of equations through a finite-difference code. These numerical integrations show the island deformation and the growth reversal which occur in the nonlinear regime and which are ascribed to a velocity shear caused by the flattening of the density profile inside the separatrix. The island deformation and the growth reversal agree qualitatively with the experimental observations on the phenomenon named ”limit cycles”. Finally in Chapter 7 we discuss our results.

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Chapter 2 Toroidal geometry and neoclassical

theory

In this section we will review the neoclassical theory, as outlined in [45] and [47], and we provide the main results which are fundamental to describe the toroidal effects on plasma rotation in a tokamak device. We examine first the geometry of magnetic field lines in a tokamak equilibrium and then we study how this affects the motion of particles in a low collisionality plasma. Neoclassical effects are fundamental to understand several phenomena taking place in toroidal plasmas, such as poloidal flow damping and the bootstrap current. These results will be useful later, when we will improve the expression for the poloidal flow damping by including the Finite Larmor Radius (FLR) corrections. The material in this chapter is presented in a form which follows very closely the book of Helander and Sigmar [45].

2.1 Magnetic field related concepts

The tokamak is an axisymmetric device, which means that all the derivatives with respect to the toroidal angle ϕ vanish for equilibrium quantities, ∂/∂ϕ = 0. In cylindrical coordinates (R, z, ϕ), where R = 0 is the symmetry axis, the magnetic field can be written as

B = ˆRBR+ ˆϕBϕ+ ˆzBz = ∇ ∧ A (2.1)

where A is the vector potential. The poloidal part of the magnetic field Bp = ˆRBR+ ˆzBz is

thus Bp = ˆR ∂Aϕ ∂z − ˆz 1 R ∂(RAϕ) ∂R = ∇ϕ ∧ ∇ψ (2.2) where we have used the expression for the curl of a vector in cylindrical coordinates, and introduced the so-called poloidal flux function

ψ(R, z) ≡ −RAϕ(R, z) (2.3)

∇ϕ is the contravariant basis vector in the direction ˆϕ, such that ˆϕ = R∇ϕ. It is clear that ψ is constant along magnetic field lines, B · ∇ψ = 0. The magnetic field vector thus lies on surfaces of constant ψ. In a tokamak they usually form nested toroids and are called flux surfaces. The

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innermost flux surface, which is just a circular line, is called the magnetic axis. The poloidal magnetic field is mainly produced by the toroidal plasma current. The toroidal part of the magnetic field is in the direction of ∇ϕ and can therefore be written as

Bt= ˆϕBϕ = I(R, z)∇ϕ (2.4)

The function I(R, z) thus defined is related to the plasma current J by Ampere’s law

µ0J = ∇ ∧ B = ∇I ∧ ∇ϕ + ∇2ψ∇ϕ + (∇ψ · ∇)∇ϕ (2.5)

Since the vectors ∇2ψ∇ϕ and (∇ψ · ∇)∇ϕ are both in the toroidal direction, we see that the poloidal part of the plasma current

Jp = µ−10 ∇I ∧ ∇ϕ (2.6)

is perpendicular to ∇I. Usually electric current cannot flow across flux surfaces, that is J ·∇ψ = 0, as follows from the equilibrium condition J ∧ B = ∇P if the pressure P is a flux function. Then ∇I ∧ ∇ψ = 0, so the function I(R, z) must be constant on flux surfaces, I(R, z) = I(ψ). Thus the expression Eq.2.1 for the magnetic field becomes

B = I(ψ)∇ϕ + ∇ϕ ∧ ∇ψ (2.7) This is generally the most convenient way of writing an axisymmetric magnetic field. An alternative form for the magnetic field can be found by first introducing a few useful geometric concepts. Let us introduce a poloidal angle coordinate θ to measure the position on a flux surface. We require θ to have a period of 2π and to vary only in the poloidal plane, ∇θ ·∇ϕ = 0, but we leave its precise definition otherwise arbitrary. The volume element becomes

dV = g1/2dψdθdϕ (2.8) where g1/2 = |(∇ϕ ∧ ∇ψ) · ∇θ|−1 = |B · ∇θ|−1 is the Jacobian. The flux surface element therefore becomes

dS = |∇ψ|

|B · ∇θ|dθdϕ (2.9)

and the line element in the poloidal direction is

dlp =

dS Rdϕ =

Bpdθ

|B · ∇θ| (2.10)

since Bp = |∇ψ|/R. A very useful concept is that of the flux-surface average. Given a

quan-tity Q(ψ, θ), its flux-surface average hQi is defined as the volume average of Q between two neighboring flux surfaces ψ and ψ + dψ

hQi (ψ) ≡ I _{Q(ψ, θ)} B · ∇θdθ .I dθ B · ∇θ (2.11) or equivalently hQi (ψ) ≡ I Qdlp Bp .I dl_p Bp (2.12)

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whereH indicates that the integration should be taken one turn around the torus in the poloidal direction. The flux-surface average annihilates the operator B · ∇ = g−1/2∂/∂θ, that is

hB · ∇f (ψ, θ)i = 0 (2.13)

for any periodic function f (ψ, θ). Using the flux-surface average, we define the safety factor as the average change in ϕ divided by the average change in θ along the magnetic field

q(ψ) ≡ hB · ∇ϕi

hB · ∇θi (2.14)

It can be interpreted as the average number of toroidal turns the magnetic field makes in one turn around the poloidal direction since

B · ∇ϕ B · ∇θ =

dϕ

dθ (2.15)

In a tokamak, the safety factor is usually of order unity or larger for stability. By using the safety factor and the poloidal angle we introduced above, the magnetic field can be written in the following form:

B = ∇ψ ∧ ∇(qθ − ϕ) (2.16)

2.2 MHD equilibrium

Now that we have introduced the basis concepts of the magnetic field in a tokamak, we consider the equilibrium which establishes between the magnetic force acting on the plasma and the pressure gradient. It is important to determine the equilibrium configuration of a tokamak because the shape of the magnetic flux surfaces determines the dynamics of the plasma and the orbits of the particles. If we add the momentum equation of all species in plasma, which we will see more in details later, we can write the force balance as

J ∧ B = ∇P (2.17)

where inertial forces and viscosity are neglected. Viscosity, which comes from the divergence of the stress tensor, is smaller than pressure in any plasma close to local thermodynamic equilibrium, and the inertial term is small if the flow velocity is smaller than thermal, nama(Va·

∇)Va ∇Pa if Va vtha. Eq.2.17 describes force balance and indicates that a plasma can

be confined by a magnetic field if plasma pressure is balanced by a magnetic force. It follows that, if this is the case, no current can flow across flux surfaces, J · ∇ψ = 0, as assumed in the previous section. Furthermore, the pressure is constant on flux surfaces, P = P (ψ), since

B · ∇P = 0 (2.18)

We remind that the time derivative was neglected in the force balance Eq.2.17, which is thus only valid on sufficiently long time scales. We are, for instance, considering phenomena that are slower than pressure equilibration along magnetic field lines. This concept will be explained

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in detail later. Multiplying the force balance relation Eq.2.17 by ∇ψ and using the form Eq.2.7 for the magnetic field, gives

[I(ψ)J ∧ ∇ϕ − (J · ∇ϕ)∇ψ] · ∇ψ = ∇P · ∇ψ (2.19) and by using Eq.2.6 for the poloidal component of the current we obtain the toroidal component of the current as J · ∇ϕ = − P0 + II 0 µ0R2 (2.20) where a prime denotes differentiation with respect to ψ. If we regard this as an equation for I(ψ) = RBϕ, we note that the variation of this quantity is driven by two sources: the pressure

gradient and the plasma current. The former tends to make the plasma diamagnetic by making I0 have opposite sign from P0. The toroidal plasma current has the opposite effect, producing paramagnetism. Inserting Eq.2.20 in the toroidal component of Ampere’s law

∇ϕ · (∇ ∧ B) = µ0∇J · ∇ϕ (2.21)

recalling the vector algebra rule ∇ · (B ∧ A) = A · (∇ ∧ B) − B · (∇ ∧ A), and writing the left-hand side of this equation as

∇ · (B ∧ ∇ϕ) = ∇ · [(∇ϕ ∧ ∇ψ) ∧ ∇ϕ] = ∇ · (R−2∇ψ) (2.22) gives the so-called Grad-Shafranov equation [67]

R2∇ · ∇ψ R2 = −µ0R2 dP dψ − I dI dψ (2.23)

whose solutions describe the possible plasma equilibria. The operator on the left is often denoted by ∇∗ψ and is explicitly equal to

∇∗ψ = R2∇ · (R−2∇ψ) = R ∂ ∂R 1 R ∂ψ ∂R + ∂ 2_ψ ∂z2 (2.24)

In analytical transport calculations, one often simplifies the analysis by assuming that the inverse aspect ratio is small, that is

= r/R0 1 (2.25)

Here r = p(R − R0)2+ z2 is called the minor radius, as opposed to the major radius R, and

R0 is the distance of the magnetic axis from the symmetry axis. Moreover, if the outermost

flux surface (at the plasma boundary) is shaped as a circle then, for small Shafranov shift, the inner flux surfaces will remain approximately circular. If the gradients of P (ψ) and I2_{(ψ) are}

not constant, it can be shown that if the plasma pressure is much smaller than the magnetic field pressure, satisfying the tokamak ordering

β ≡ P B2_/2µ

0

≈ 2 ₁ _(2.26)

there are equilibria with nearly circular cross section, and with the safety factor (Eq.2.14) of order unity. The outer magnetic surfaces are slightly shifted from the magnetic axis towards

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the inside of the tokamak with a a Shafranov shift of order ∆ ≈ r. When flux surfaces are circular, the natural candidate for the poloidal angle θ is the usual poloidal angle. If we solve the equilibrium in this approximation we find the usual θ-dependence of the toroidal magnetic field

Bϕ(θ) = B0(1 − cos θ) (2.27)

where B0 ≡ R0I(ψ) and the safety factor Eq.2.14 becomes

q(r) = rB0 RBθ0(r)

(2.28)

2.3 Particle orbits

The motion of a charged particle in a strong, stationary, axisymmetric magnetic field is char-acterized by three constants of motion,

E = 1 2mv 2_{+ Zeφ} µ = mv 2 ⊥ 2B pϕ = mRvϕ− Zeψ (2.29)

Here the magnetic field is assumed to be strong enough to make the gyroradius ρ = v⊥/Ω

significantly smaller than the characteristic length scale of the field, so that the magnetic moment µ is conserved. The third invariant pϕ stems from axisymmetry.

Since each of these particles stays close to a particular flux surface, the velocity v remains approximately constant. It is then convenient to describe the orbits by the three approximate constants of motion (v, λ, ψ), where

λ ≡ µB0 mv2_/2 =

v2 ⊥B0

v2_B (2.30)

rather than the exact ones (E, µ, pϕ). The λ parameter is related to the pitch angle α, that is

the angle between the velocity vector of the particles and the magnetic field, by λ = h sin2α, where

h ≡ B0

B (2.31)

is the toroidal metric coefficient. B0 is the value of the magnetic field on axis.

Now let us consider in more details orbits of particles moving on a flux surface where the magnetic field varies between Bmin on the outboard side and Bmax on the inboard side of the

surface, according to Eq.2.27. In general, a particle with given energy E and magnetic moment µ can never enter a region where the magnetic field is so strong that µB > E − Zeφ, since then v⊥ > v and v_k2 = 2(E − Zeφ − µB)/m < 0, which is unphysical. If φ is constant along the

orbit, so that v is conserved, this means that λ always stays below B0/B. Thus, while all the

particles must satisfy

0 ≤ λ ≤ B0 Bmin

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the ones with B0 Bmax < λ ≤ B0 Bmin (2.33) can only lie on the outboard side of the flux surface. When one of the latter particles moves along a field line towards the inside, it is reflected by the mirror force Fk = −µ∇kB, so that vk

changes sign. Such particles are referred to as ”trapped” (on the outside of the torus). Particles with 0 < λ < B0/Bmaxare free to move along field lines and explore the entire flux surface. The

mirror force slows them down as they approach the strong magnetic field on the inside, but it is not strong enough to reflect them. Such particles are referred to as ”passing” or ”circulating” (around the torus).

circulating

trapped

Z

R

Figure 2.1: Banana orbits

In a large-aspect-ratio tokamak with circular flux surfaces and q ≈ O(1), the magnetic field is dominated by its toroidal component, so that B ≈ Bϕ. Its strength varies between

Bmin = B0/(1 + ) on the outside and Bmax = B0/(1 − ) on the inside of the torus, and

the particle orbits can thus be classified as circulating, if 0 ≤ λ < 1 − , and trapped, if 1 − < λ ≤ 1 + . We see that the trapped particles constitute a small fraction, O(√) 1, of the total number of particles if 1. Because of their shape, the trapped orbits are also called banana orbits (Fig.2.1). The parallel velocity of a particle is

vk = σv

p

1 − λB/B0 ≈ σv

p

1 − λ(1 − cos θ) (2.34)

where σ is the sign of vk. The particles deviate from the motion along the flux surfaces due to

the drift motion. The typical radial excursion of a trapped particle in its motion along a banana orbit is the banana width, and is the characteristic step size of the neoclassical transport:

∆r = ∆ψ RBp

= ∆vk Ωp

≈√ρp (2.35)

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orbit (often called the bounce time) is equal to τb = I dθ vkb · ∇θ (2.36)

The integral is to be taken one poloidal turn around the orbit. For a circulating particle, this means that the integral is taken around the flux surface 0 ≤ θ ≤ 2π, while for trapped particles it follows a banana orbit. It is useful to introduce the trapping parameter

k2 = 1 − λ(1 − )

2λ (2.37)

so that 0 < k < 1 for trapped particles and k > 1 for circulating ones. For trapped particles k is related to the poloidal bounce angle θb by k = sin θb/2. The bounce angle is the poloidal

location of the turning point, such that vk(θb) = 0. We may now express the bounce time as

τb = qR v√2λ I dθ σpk2_{− sin}2_θ/2 (2.38)

Particle motion in a large-aspect-ratio tokamak is entirely analogous to the motion of an ordi-nary pendulum. The angle of the pendulum to the vertical can be identified with the poloidal angle θ. Trapped motion corresponds to the pendulum executing finite oscillations, and cir-culating motion corresponds to the pendulum swinging all the way over the top around its pivot. The expression for the bounce time derived above is similar to the period of a pendulum. In particular, it tends to infinity if the turning point of the pendulum approaches the upper, unstable equilibrium point.

2.4 Transport ordering

Transport theory relies on a particular ordering of the relative importance of various physical processes in the plasma. The fundamental expansion parameter is that of the smallness of the Larmor radius ρ compared with the macroscopic scale length L,

δ ≡ ρ/L 1 (2.39)

The requirement that this parameter be small must hold for all particle species, in particular for the ions because the ion Larmor radius is much larger than the electrons. Gradients (of density, temperature, magnetic field etc.) are assumed to be small , so that they cause small field variations on the scale of ρ. The task of transport theory is to describe the rates of relaxation of these gradients towards global thermodynamic equilibrium. In fact, gradients in the equilibrium quantities such as density, temperature and electrostatic potential, cause fluxes which tend to reduce these gradients. As long as the step length in the random walk is smaller than the gradient length scale this process is diffusive, so the associated time derivative is expected to be of order ∂ ∂t ≈ D L2 ≈ δ 2_ν _(2.40)

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where ν is the collision frequency and D ≈ νρ2 _{the cross-field diffusion coefficient. As usual,}

we shall assume that the plasma is magnetized in the sense that the gyrofrequency exceeds the collision frequency,

∆ ≡ ν

Ω 1 (2.41)

a requirement that in tokamaks is practically always satisfied with a wide margin. We note that we can also express ∆ as the ratio between the Larmor radius ρ = vth/Ω and the mean

free path λ = vth/ν,

∆ = ρ

λ (2.42)

so the condition ∆ 1 again expresses the smallness of the Larmor radius, and ∆ = O(δ) if the mean free path is comparable to the macroscopic length scale L. The requirement ∆ = O(δ) implies that the time derivative is considerably smaller than the transit frequency,

∂ ∂t ≈ δ

2vth

L (2.43)

Finally, realistic flow velocities are smaller than the thermal speed,

V ≈ δvth (2.44)

Sometimes is useful to introduce the large-aspect-ratio approximation, that is = r/R 1. To be consistent with this ordering, the requirement that the plasma is magnetized must be interpreted in the stronger sense that the poloidal Larmor radius ρp is smaller than the minor

radius, ρp/r 1. In the following we will assume this implicitly.

2.5 Collisionality

The physics of neoclassical transport depends on the relative magnitude of the collision fre-quency ν and the transit frefre-quency ωt = vth/(qR), the so-called collisionality. It is similar

for electrons and ions since νee/νii ≈ vthe/vthi but it may differ among ion species with very

disparate masses. If the collisionality is large, L λ ≈

ν vth/(qR)

1 (2.45)

the mean free path λ = vth/ν is shorter than the parallel distance around a flux surface L ≈ qR

(connection length) and the Braginskii fluid equations may be applied for the analysis. In fact Braginskii, in his work ”Transport Processes in a Plasma” [8] deals with collisional plasmas, where the effects of the inhomogeneity of the magnetic field due to the toroidal geometry don’t affect the particle motion. Particles in this regime of collisionality do not fully complete a bounce orbit because their motion is disturbed by collisions before an orbit has been completed. This high collisionality regime is called Pfirsch-Schl¨uter, or fluid regime [58]. In the opposite limit,

ν vth/(qR)

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referred to as banana-plateau regime, orbits are completed and short mean free path closure of the fluid equations is inapplicable. The core of the tokamak, where temperature is higher, is usually in this regime. If the aspect ratio is large, 1, the banana-plateau regime is subdivided into two regimes: the plateau regime

3/2 ν vth/(qR)

1 (2.47)

and the banana regime

ν vth/(qR)

3/2 _(2.48)

In the former, most circulating particle orbits are completed but trapped orbits are destroyed by collisions before completion since the effective collision frequency, νef f = ν/, required to

scatter a trapped particle out of its magnetic well, ∆B/B ≈ , is larger than the bounce frequency ωb ≈ √ vth/(qR), that is ν∗ ≡ ν/ ωb = ν/ 3/2 vth/(qR) 1 (2.49)

At large aspect ratio, transport is quite different in different collisionality regimes. The diffu-sivity of particles and heat is proportional to the collision frequency in the banana and Pfirsch-Schl¨uter regimes (with different proportionality constants) but is independent of collisionality in the plateau regime (Fig.2.2).

D νRq vth banana plateau Pfirsc h-Sc hl¨uter 3/2 ₁

Figure 2.2: Dependence of D on the collision frequency

2.6 The kinetic equation

A plasma is made of many charged particles, in particular electrons, primary ions and impurity ions, each of which is characterized by its position r and velocity v. Thus each particle can be represented as a point in a six-dimensional phase space (r, v). Due to the large number of particles constituting a plasma, a single-particle approach is impossible. Therefore a statistical approach in needed. The distribution function fa(r, v, t) of a particle species a is defined as the

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number of particles of that species per unit volume near the point z = (r, v) in phase space at time t. That is, the quantity

fa(r, v, t)d3rd3v (2.50)

is the number of particles in the volume element d3rd3v about the point (r, v). Thus if we integrate fa over velocity space, we obtain the number density of particles in the real space,

Z

fa(r, v, t)d3v = na(r, t) (2.51)

In a plasma, each particle moves according to the equations of motion ˙r = v

˙v = ea ma

(E + v ∧ B) (2.52)

where ea is the electric charge of the particle and ma its mass. The time evolution of the

distribution function fa in the phase space can be described in terms of the flow induced by

the six-dimensional vector ˙z = ( ˙r, ˙v): ∂fa

∂t + ∂

∂z( ˙zfa) = 0 (2.53) Close to each particle, electric and magnetic fields strongly fluctuate and they are dominant with respect to the mean fields, generated by all the other particles. These small-scale fluctuations become negligible for distances much larger than the Debye length, so that for a macroscopic description of the plasma we can ignore them and consider E and B as mean fields. The effect of collisions on the distribution function is taken into account in the collision operator,

Ca(fa) = ∂fa ∂t coll (2.54)

This term is usually put on the right-hand side of the kinetic equation, ∂fa ∂t + v · ∇fa+ ea ma (E + v ∧ B)∂fa ∂v = Ca(fa) (2.55) When collisions are negligible, Eq.2.55 becomes the Vlasov equation. In a neutral gas, Ca

is the Boltzmann operator and it describes the hard-sphere elastic collisions between neutral particles. In a plasma, the charged particles interact by Coulomb force, so Ca is called the

Coulomb operator. The collision operator

Ca =

X

b

Cab(fa, fb) (2.56)

is the sum of the collisions between the species a particles with every other species b particles. It must satisfy certain conservation laws. The conservation of particles is expressed by

Z

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The conservation of momentum and energy become Z mavCab(fa)d3v = − Z mbvCba(fb)d3v Z _m av2 2 Cab(fa)d 3_{v = −} Z _m bv2 2 Cba(fb)d 3_v (2.58)

The first of Eq.2.58 means that the force exerted by particle a on particle b is equal and opposite to that exerted by b on a. The second of Eq.2.58 means that no energy is produced by collisions. Another important property of the collision operator is that it drives the distribution function towards thermodynamical equilibrium, that is towards a shifted Maxwellian,

fM a(r, t) = na(r, t) ma 2πTa 3/2 exp −ma[v − Va] 2 2Ta (2.59)

where Va is the mean (fluid) velocity and Ta the temperature.

2.7 Fluid equations

Kinetic theory provides an accurate statistical description of plasma dynamics. However, for many applications to plasma physics, it is not necessary to solve the kinetic problem. More useful information from the distribution function fa is contained in its first few moments, that

is integrals over velocity space of fa multiplied by different functions of v. The zeroth order

moment is the density

na(r, t) =

Z

fa(r, v, t)d3v (2.60)

The macroscopic fluid velocity is defined by

Va(r, t) = hvi_f ≡

1 na

Z

vfa(r, v, t)d3v (2.61)

and is equal to the average velocity of all particles of species a in a certain point in space. The velocity of a particular particle differs from this average velocity by v0_a ≡ v − Va. The

temperature Ta is defined so that 3Ta/2 represents the average kinetic energy associated with

these random velocities,

3 2Ta= ma(va0)2 2 f (2.62)

If we take the mav2/2 moment of the distribution function,

manahv2i_f 2 = manaV2a 2 + 3naTa 2 (2.63)

we find that the total energy is the sum of the kinetic energy associated with the mean flow V and the thermal energy. The first two moment equations are thus (neglecting particle sources)

∂n

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∂mnV

∂t + ∇ · Π = ne(E + V ∧ B) + F (2.65) where we dropped the species index and introduced the stress tensor

Π ≡ hmnvvi_f (2.66)

The right hand side of Eq.2.65 contains the rate of change of momentum due to the effect of electric field and collisional friction. The momentum exchange due to collisional friction is given by

F ≡ Z

mvC(f )d3v (2.67) The stress tensor Π can be written in a more useful form,

Π = P I + π + mnV V (2.68) With this notation, the first two moment equations can be written as

dna

dt + na∇ · Va = 0 (2.69) mana

dVa

dt = −∇Pa− ∇ · πa+ naea(E + Va∧ B) + Fa (2.70) where we introduced the total derivative,

d dt ≡

∂

∂t+ Va· ∇ (2.71) This reminder on fluid equations had the purpose of introducing a few macroscopic quantities, such as the particle density, the fluid velocity and, most importantly, the stress tensor (which we will use to express the relation between fluxes and viscosity).

2.8 The drift-kinetic equation

To study neoclassical phenomena, it is more suitable to use the guiding centre variables w = (R, E, µ, θ), where R is the position of the guiding-centre of the particle, E is the total energy, µ is the magnetic moment (see Eq.2.29) and θ is the gyro-angle of the particle in its rotation around the magnetic field lines. Further details on the derivation of the drift-kinetic equation can be found in the fundamental paper by Hinton & Hazeltine ”Theory of plasma transport in toroidal confinement systems” [46] or in the book ”Plasma confinement” by Hazeltine & Meiss [44]. Using the phase-space variables w, Eq.2.55 becomes

∂fa ∂t + ˙R · ∇fa+ ˙E ∂fa ∂E + ˙µ ∂fa ∂µ + ˙θ ∂fa ∂θ = Ca(fa) (2.72) Here we are interested in phenomena characterized by frequencies which are much smaller than the Larmor frequency Ω and length-scales much larger than the ion Larmor radius ρ. We can thus use the small parameters δ = ρ/L and ∆ = ν/Ω, which we have previously defined.

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According to the transport ordering, ∂/∂t ≈ δ2_{ν, so this term is negligible. The second term,}

˙

R · ∇fa, can be decomposed in a contribution from the parallel velocity vk and a contribution

from the drift velocity vd:

˙

R · ∇fa= vk∇kfa+ vd· ∇fa (2.73)

The drift velocity is typically smaller than the parallel velocity by a factor δ, which is not compensated by the perpendicular gradient. In fact the parallel gradient scales as the tokamak parallel length scale, which is proportional to the major radius R0, while the perpendicular

gradient usually scales as the minor radius a. The dimensionless quantity a/R0 = is the inverse

aspect ratio, and it is usually small in a tokamak. However, we are ordering the quantities in the small parameter δ previously defined, which is much smaller than in all cases. So the drift term is negligible to lowest order in δ. The parallel velocity is of the order of the thermal velocity vth, which can be written as vth≈ ρΩ. So we have

vk∇kfa ≈

ρ

LΩfa (2.74)

The collision operator is ordered with ν, according to the simplified BGK (Bhatnagar-Gross-Krook) model [5]

C(fa) ≈ ν(f0a− fa) (2.75)

so the ratio between the convective term ˙R · ∇fa and the collision term is

vk∇kfa Ca(fa) ≈ ρ L Ω ν = δ ∆ (2.76)

Since ∆ = O(δ) if the parallel mean free path is of the order of the parallel length scale, these two terms are approximately of the same order. The magnetic moment µ and the energy E are constant for all our purposes, so the third and fourth terms of Eq.2.72 are zero. The last term on the left of Eq.2.72 describes Larmor rotation, because ˙θ = Ω, and it is usually the largest term in the equation. Therefore, to the zeroth order in δ and ∆, the distribution function is independent on the gyroangle

Ω∂f0a

∂θ = 0 (2.77)

An equation for f0a(R, , µ, t) is obtained by taking the average over the gyroangle θ, with the

result

vk∇kf0a = Ca(f0a) (2.78)

We are going to show here that Eq.2.78 has as a solution the Maxwellian distribution. This derivation can be found in [47] or in [46]. To do that we multiply this equation by log f0a,

integrate over velocity space and apply the flux-surface average. This procedure annihilates the left hand side and we are left with

Z

log f0aCa(f0a)d3v

= 0 (2.79)

for each particle species. For the ions in a simple plasma, collisions with electrons are weak, Cie/Cii≈ (me/mi)1/2 1, so Ci ≈ Cii. According to the H-theorem (Boltzmann, 1872)

Z

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with the equality if and only if f is Maxwellian. We conclude that to lowest order in δ the ion distribution must be Maxwellian,

fi0 = fM i(ψ) = ni(ψ) mi 2πTi(ψ) 3/2 exp −miv 2_{/2 + Zeφ} Ti(ψ) (2.81)

For the electrons Ce= Cee+ Cei and in the ion rest frame

Z

log fe0Cei(fe0)d3v ≤ 0 (2.82)

The electrons are therefore Maxwellian with the same flow velocity as the ions in lowest order. We have thus proved that the lowest order drift-kinetic equation Eq.2.78 only has Maxwellian solutions that are constant on flux surfaces. In the case of low collisionality, the rapid parallel streaming caused by vk∇kf0a establishes a distribution that is constant on each flux surface,

∇kf0a = 0, in a bounce time, while collisions drive this distribution towards a Maxwellian in a

collision time. On the other hand, if the collisionality is high, local thermodynamic equilibrium is first established at each point in the plasma in a collision time. Then, on the longer time scale associated with parallel collisional transport, the density and temperature equilibrate on each flux surface. Regardless of the order in which these processes occur, the equilibration over the flux surfaces takes a very short time compared to the transport processes.

The first order drift-kinetic equation is

vk∇kfa1+ vd· ∇fa0+ eavkE_k(A)

∂fa0

∂E = Ca(fa1) (2.83) where E_k(A) is the parallel inductive electric field, because we assumed that the parallel gradient of the electrostatic potential is zero. Eq.2.83 determines the correction fa1 to the lowest-order

Maxwellian guiding centre distribution fa0. Here and in the rest of the document, Ca denotes

the linearized collision operator of species a. The term vd·∇fa0 = hvd· ∇ψi ∂fa0/∂ψ represents

the effect of the cross-field drift, that is the radial particle flux across the magnetic surfaces, which can be written as

˙ R · ∇ψ = vk(b ∧ ∇ψ) · ∇ vk Ωa = Ivk∇k vk Ωa (2.84)

with the derivative ∇k taken at constant E and µ. To arrive at Eq.2.84 we have used RBϕ = I

and

B ∧ ∇ψ B2 =

I

Bb − R ˆϕ (2.85) Using the expression Eq.2.84, the first-order drift-kinetic equation becomes

vk∇k fa1+ Ivk Ωa ∂fa0 ∂ψ − eaE (A) k Ta vkfa0= Ca(fa1) (2.86)

The task of neoclassical theory is essentially to solve this equation in various collisionality regimes.

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2.9 Parallel particle flux

We can distinguish three different kinds of plasma motions, namely the parallel flow, which is parallel to the magnetic field lines, the first-order flow, which is perpendicular to the magnetic field lines but lies on the magnetic surfaces, and finally the flux across the magnetic surfaces (cross-field) [14]. The parallel flow is ordered with the thermal velocity vth, the first-order

flow is smaller than vth by a factor δ and the cross-field flux by a factor δ2. These motions

are associated with three different dynamics: the parallel flow is associated with the thermal equilibration along the magnetic field lines, the first-order flow is associated to the drift motions which take place over the magnetic surfaces and the cross flux is the slow diffusion of particles across the magnetic surfaces. In a cylindrical plasma the first-order flows of heat and particles, due to the diamagnetic and E ∧ B drift motions, are divergence-free. In a tokamak, on the contrary, the inhomogeneity of the magnetic field causes the first-order flow to acquire a non-zero divergence. This causes, as we will see, the appearance of a parallel flux whose parallel gradient compensates the divergence of the first-order flow. The perpendicular component of the flow in lowest order equals the diamagnetic flow, which we write as

naVa⊥= na(ψ)ωa(ψ) R ˆϕ − I Bb (2.87) where ωa(ψ) ≡ − dφ dψ − 1 naea dPa dψ (2.88)

Note that to lowest order Pa and φ are flux functions, so the flow is perpendicular to ∇ψ and

thus tangential to the flux surface. Eq.2.87 represents the perpendicular component of the flow velocity within the flux surface and is of order V ≈ δvth. The parallel component can be

determined by observing that the total flow within the flux surface must be divergence free to zeroth order in δ, otherwise the continuity equation

∂n

∂t + ∇ · (nV ) = 0 (2.89) would imply larger time derivatives,

∂ log n ∂t ≈ V L ≈ δ vth L (2.90)

than allowed by the transport ordering. However, even in an axisymmetric field (∂/∂ϕ = 0), the diamagnetic flow is not itself divergence free, and thus it must be balanced by a parallel ”return flow” Vak with

∇ · (naVak) = −∇ · (naVa⊥) (2.91)

By taking explicitly the divergence, we find the following form for the parallel flow

naVak =

1

BI(ψ)na(ψ)ωa(ψ) + Ka(ψ)B (2.92) where Ka(ψ) is an integration constant. The total flow within a flux surface thus becomes

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The first term describes purely toroidal rotation of the species a. The angular frequency ωa

may differ from surface to surface and from species to species, but within every flux surface each species fluid rotates as a rigid body. The poloidal rotation is entirely contained in the second term, which is parallel to the magnetic field. The heat flux can be treated similarly.

2.10 Flow across flux surfaces

Applying Gauss’ law to the continuity equation Eq.2.89 gives

− ∂ ∂t Z V nad3r = Z ∂V naVa· dS (2.94)

Let us take the first integral over the volume V inside some flux surface ψ. Since the surface element is dS = ∇ψdθdϕ/B · ∇θ, we conclude that the particle flux across the flux surface is

− ∂ ∂t Z V nad3r = hnaVa· ∇ψi I _dθ B · ∇θ (2.95) One aim of transport theory is to calculate this quantity, which apart from a geometrical factor thus equals hnaVa· ∇ψi, or

hΓa· ∇ψi = hR ˆϕ · (naVa∧ B)i (2.96)

where Γa ≡ naVa and we have used the representation Eq.2.7 for the magnetic field. This

form of the cross-field flux suggests that we use the flux-surface averaged R ˆϕ-projection of the momentum equation Eq.2.65 to write

heaΓa· ∇ψi = R ˆϕ · ∂manaVa ∂t + hR ˆϕ · ∇ · Πi −DnaeaRE(A)ϕ E − hRFaϕi (2.97)

Note that the pressure gradient does not appear because R ˆϕ · ∇Pa= ∂Pa/∂ϕ = 0 by

axisym-metry. The first two terms on the right are small in transport theory so we have heaΓa· ∇ψi = −

D

naeaRE(A)ϕ

E

− hRFaϕi (2.98)

That the hR ˆϕ · ∇ · Πi component of the viscosity is small implies weak radial transport of angular momentum, and has important consequences for the evolution of the radial electric field. It is useful to decompose the cross-field flux Eq.2.98 into three terms,

hΓa· ∇ψi = hΓa· ∇ψicl+ * na E(A)∧ B B2 · ∇ψ + + hΓa· ∇ψineo (2.99)

The first is the classical flux,

hΓa· ∇ψi cl

≡ hRFa⊥ϕ/eai (2.100)

which is driven by perpendicular friction. The second term is the E ∧ B drift across the flux surface, and the remainder is defined to be the neoclassical flux. It can be written as

hΓa· ∇ψi neo ≡ −I* Fk+ naeaE (A) k eaB + (2.101)

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by using Eq.2.85. Thus, while the classical flux is associated with perpendicular friction, the neoclassical flux has to do with parallel friction. This reflects the different mechanisms of these transport processes. Classical transport is caused by collisional relaxation of the diamagnetic flow associated with gyromotion, which is perpendicular to the magnetic field. Neoclassical transport is caused by friction acting on guiding centres, which predominantly move along the field. In a torus with large aspect ratio the neoclassical flux is usually much larger than the classical flux. Both the classical and the neoclassical flux are automatically ambipolar. That is, regardless of any radial electric field, we always have

X

a

eahΓa· ∇ψi = 0 (2.102)

because of momentum conservation P

aFa = 0 and quasi-neutrality,

P

anaea = 0. This

ambipolarity property is a consequence of the transport ordering we have adopted. The neoclassical flux can be obtained kinetically as

hΓa· ∇ψineo =

Z

favd· ∇ψd3v

(2.103)

Neoclassical transport is thus driven by the deviation of the guiding-centre distribution from a Maxwellian. Classical transport is caused by the difference between the distribution function of particles fa(r) and guiding centres fa(R), that is by the departure from a Maxwellian caused

by finite Larmor radius. The radial heat flux across flux surfaces is obtained similarly.

2.11 Transport in the banana regime

We are interested in describing the dynamics of magnetic islands in a plasma with low collision-ality, where the effect of banana orbits on plasma dynamics is important, so we limit ourselves to study the banana regime. In this regime the effective collision frequency for scattering of trapped particles is much smaller than the bounce frequency of these particles:

ν∗ =

νef f

ωb

1 (2.104)

and we must deal with the full drift-kinetic equation Eq.2.86. We recall that the expansion fa = fa0 + fa1+ · · · refers to the smallness of the Larmor radius, and that the gradients are

taken at fixed energy E = mav2/2 + eaφ and magnetic moment µ = mav⊥2/(2B), so that the

mirror force is accounted for by the parallel gradient. For each species fa0 is a Maxwellian,

and the drift-kinetic equation Eq.2.86 is to be solved for fa1. It is clear that there are two

driving terms. The first one is the cross-field drift, vd· ∇fa0, which is proportional to the radial

gradients in ∂fa0/∂ψ, and the second one is associated with the induced electric field E (A) k . The

latter can be conveniently accounted for by using the Spitzer function fas defined by

Ca(fas) = −

eavkE_k(A)

Ta

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The Spitzer problem has been solved accurately in the literature [76], thus we can regard fas

as a known function and write the drift-kinetic equation as

vk∇k(fa1− Fa) = Ca(fa1− fas) (2.106) where Fa ≡ − Ivk Ωa ∂fa0 ∂ψ = − Ivk Ωa ∂ log na ∂ψ + ea Ta ∂φ ∂ψ + mav2 2Ta − 3 2 ∂ log Ta ∂ψ fa0 (2.107)

To proceed analytically from here we use the banana regime assumption Eq.2.104 of low colli-sionality. Accordingly, we make a subsidiary expansion of fa1 in the smallness of the collision

frequency, fa1= f (0) a1 + f

(1)

a1 + · · · , and try to solve

vk∇k(fa1(0)− Fa) = 0 vk∇kf (1) a1 = Ca(f (0) a1 − fas) (2.108)

The first of these equations is solved by

f_a1(0) = ga+ Fa (2.109)

where ga is an integration constant independent of θ, that is ∇kga = 0 and ga= ga(ψ, E.µ, σ).

For passing particles, the function ga can be determined from the solubility constraint obtained

by multiplying the next-order equation Eq.2.108 by B/vk and taking the flux-surface average,

B vk

Ca(ga+ Fa− fas)

= 0 (2.110)

for passing particles. While ga is determined by Eq.2.110 in the circulating domain, it vanishes

in the trapped particle space,

ga = 0 (2.111)

for trapped particles. We are not going to demonstrate this statement because the proof can be found in the literature [45], [46]. So the trapped particle response to the radial gradients is entirely described by fa1 = Fa. The distribution function of passing particles is determined

by the collisional equilibrium they establish with the trapped population. Mathematically, our remaining task is to solve Eq.2.110 for ga in the passing region, which is done separately for

ions and electrons. For electrons, the source term associated with the electric field is generally of the same order as that associated with radial gradients, that is

fes≈ Fe (2.112)

Since the latter is derived from the cross-field drift, it is proportional to the Larmor radius, and is therefore much larger for the ions. It will therefore dominate over fis, which does not scale

with gyroradius,

fis Fi (2.113)

In the ion equation we may therefore neglect the parallel electric field. The parallel particle flux associated with Fa,

Z

vkFad3v =

Inaωa

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where the frequency ωa, previously defined in Eq.2.88, has the character of a diamagnetic flow

in the following sense. If we consider the banana orbits that pass through a particular point in the outer midplane, those with vk > 0 have their bounce points closer to the centre of the

plasma and those with vk < 0 have their bounce points further out. If the density is centrally

peaked, the former are more numerous, and hence a parallel diamagnetic flow is produced (see Fig.2.3). Unlike the usual diamagnetic flow, which has to do with Larmor orbits, this flow is associated with banana orbits.

From general arguments it has been earlier established that the parallel flow velocity of each species is given by Eq.2.92 and the total flow by Eq.2.93, regardless of the collisionality. It is then clear from Eq.2.114 that, in the banana regime, poloidal rotation is entirely contained in ga and fas, Ka = 1 B Z (ga+ fas)vkd3v (2.115)

For ions, only the first term matters in the ordering we have adopted, and since gi vanishes in

the trapped region, we conclude that only passing ions contribute to the poloidal rotation of the plasma. This is natural since trapped particles are locked in the magnetic well.

Z R f lux surf ace low density high density

Figure 2.3: Diamagnetic effects

2.12 Ion transport

We now calculate the transport of ions by solving the ion version of Eq.2.110, where ion-electron collisions are negligible, Ci ≈ Cii, and we may also neglect the parallel electric field

by disregarding fis. Proceeding to find this solution approximately, we employ the following

model operator for ion-ion collisions,

Cii(fi) = νDii(v) L(fi1) + mivkui Ti fi0 (2.116)

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Eq.2.116 can be deduced in the case of electron-ion collisions, where it describes the collision of electrons on stationary, infinitely heavy ions. However, it turns out that it is also suitable to describe like-particle collisions. L(fi1) is the so-called Lorentz operator, which describes

pitch-angle scattering and contains the angular part of the Laplace operator,

L(f ) = 1 2 1 sin θ ∂ ∂θ sin θ∂f ∂θ + 1 sin2θ ∂2f ∂ϕ2 (2.117)

The operator Eq.2.117 can be written in terms of the λ parameter defined in Eq.2.30

L = 2hvk v2 ∂ ∂λλvk ∂ ∂λ (2.118)

The quantity ui is introduced in Eq.2.116 to make sure that the collision operator conserves

the momentum in collisions,

ui = Z vkνDii(v)fid3v .Z ν_Dii(v)miv 2 3Ti fM id3v (2.119) The coefficient νii

D(v) is the velocity-dependent collision frequency. It can be expressed in terms

of the Rosenbluth potentials and scales approximately with v−3. The kinetic problem Eq.2.110, B vk L(gi+ Fi) + mivkui Ti fi0 = 0 (2.120)

can now be written as

∂ ∂λλvk ∂ gi ∂λ = − v2 2si(v, ψ)fi0 (2.121) where si(v, ψ) ≡ I hΩi ∂ log fi0 ∂ψ + Du_i h Em_i Ti (2.122) We have managed to reduce the original kinetic equation Eq.2.55 from a seven-dimensional partial differential equation to an ordinary differential equation by using the small-Larmor-radius approximation and the transport ordering, which removed gyromotion and time variation from the problem. Since transport along the magnetic field is faster than that across the magnetic surfaces, the transport problem could be considered at one flux surface at a time, which eliminates the radial coordinate. Furthermore, since the plasma is axisymmetric the toroidal coordinate ϕ is ignorable. Finally, the adoption of a model collision operator enabled us to ignore one of the velocity dimensions.

The ordinary differential equation Eq.2.121 is solved by integrating twice over λ. Since we already know that gi vanishes in the trapped domain, we only have to consider the passing

domain 0 ≤ λ ≤ B0/Bmax, where Bmax(ψ) is the maximum magnetic field strength on the flux

surface ψ. Furthermore, since gi is continuous, it vanishes at the trapped-passing boundary

λc≡ B0/Bmax. Hence gi = H(λc− λ)Vksifi0 (2.123) where Vk(λ, v, ψ) ≡ v2 2 Z λc λ dλ0 vk(λ0) = σv 2 Z λc λ dλ0 D p1 − λ0_/h(θ)E (2.124)

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and H is the Heaviside step function, so that H(λc−λ) = 1 in the passing region and H(λc−λ) =

0 in the trapped region. In a large aspect ratio torus the magnetic field strength is almost constant, h = B0/B = 1 + O(), and the trapped-passing boundary is located at λc = 1 − .

The quantity we have denoted by Vk is therefore approximately equal to the parallel velocity,

Vk = vk[1 + O()] in most of the velocity space, 1 − λ . In fact, if → 0 then

Vk → σv 2 Z 1 λ dλ0 √ 1 − λ0 = σv √ 1 − λ = vk (2.125)

The full ion distribution function fi1 = gi + Fi is obtained by piecing together the different

results we have obtained,

fi1= − I hΩi (hvk− HVk) ∂fi0 ∂ψ + miHVk Ti Du_i h E fi0 (2.126)

and the coefficient ui has been defined in Eq.2.119. Introducing the velocity-space average

{F (v)} ≡ Z Fmv 2 k nTa fMd3v = 8 3√π Z ∞ 0 F (x)e−x2x4dx (2.127) we can write the equation for ui as

{νii D} Du_i h E = 1 hni Z ν_Diivkfi1d3v (2.128)

To evaluate the integral on the right of this equation using the solution Eq.2.126, it is useful to note that for any function A(v),

Z A(v)mavkHVk hTa fa0d3v = (1 − ft)na{A} (2.129) so that Z A(v)mavk(hvk− HVk) hTa fa0d3v = ftna{A} (2.130)

where we have used hh−2i = 1. We have also introduced the effective fraction of trapped particles ft= 1 − 3 4 Z λc 0 λdλ p1 − λ/h ≈ 1.46 √ (2.131)

This approximation for ft holds in a circular torus with large aspect ratio. The number fc =

1 − ft is called the effective fraction of circulating particles.

Returning to the equation Eq.2.128 for ui and inserting the solution now gives

Du_i h E = − ITi mihΩi {νii D∂ log fi0/∂ψ} {νii D} ≈ − ITi mihΩi d log Pi dψ + Ze Ti dφ dψ − 1.17 d log Ti dψ (2.132) where −1.17 = {ν_Dii(x2 − 5/2)}/{νii

D}. We can now finally write our solution Eq.2.126 to ion

drift-kinetic equation as fi1 = − Ivk Ωi ∂fi0 ∂ψ + IHVk hΩi miv2 2Ti − 1.33 ∂ log Ti ∂ψ fi0 = Fi+ gi (2.133)

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where Fi has been defined in Eq.2.107. Note that gi is proportional to the ion temperature

gra-dient, as anticipated, and that the distribution function has a discontinuous derivative ∂fi1/∂λ

at the trapped-passing boundary because of the Heaviside function H(λc− λ). However, the

distribution function fi1 is continuous in λ = λc because Vk(λc) = 0.

Knowing the ion distribution function, we are now in a position to calculate quantities of phys-ical interest by taking moments of fi1. For instance, the flow velocity Eq.2.93 of the ions within

the flux surface can be calculated by using Eq.2.115, which implies

KiB2 = B₀ h Z givkd3v (2.134)

The quantity Ki determining the poloidal flow becomes

Ki = 1.17ff

niI

miΩiB0

dTi

dψ (2.135)

The poloidal flow niViθ = KiBθ is simply proportional to the temperature gradient. The

gra-dients of the pressure and electrostatic potential determine the toroidal flow velocity according to Eq.2.93 and the parallel flow becomes

Uik = − ITi miΩi h d log Pi dψ + Ze Ti dφ dψ − 1.17h−1fc d log Ti dψ (2.136)

where we chose to name it U instead of V to avoid confusion with the function Vki defined

above. We now specialize to the case of a large-aspect-ratio plasma with circular flux surfaces. Only at large aspect ratio (or high ion charge) it is possible to rigorously justify the use of the model operator Eq.2.116. If 1 so that h = 1 + O(), and if the flux surfaces are circular so that dψ = RBθdr, the parallel mean ion velocity Eq.2.136 becomes

Uik= − Ti miΩiθ d log Pi dr + Ze Ti dφ dr − 1.17 d log Ti dr (2.137)

where Ωiθ = ZeBθ/mi is the poloidal ion gyrofrequency. The poloidal rotation of the plasma

is given by Viθ = KiBθ niB = 1.17 miΩi dTi dr (2.138)

and is thus of order Viθ ≈ vthiρi/L, which is typically smaller than the parallel velocity Eq.2.137,

which is order Vik≈ vthiρiθ/L.

2.13 The parallel viscous force

We begin recalling that the cross-field particle flux can be obtained by taking the toroidal projection of the momentum equation. After splitting off the classical flux (due to perpendicular friction) and the E ∧ B flux, the neoclassical flux Eq.2.101 remains. We now decompose the latter into the Pfirsch-Schl¨uter flux and the banana-plateau flux,

Study of magnetic island propagation and evolution in tokamak plasmas

Universit´

a degli Studi di Pisa

Corso di Dottorato di Ricerca in Fisica

XXIX Ciclo (2013-2016)

PhD Thesis

STUDY OF MAGNETIC ISLAND

PROPAGATION AND EVOLUTION IN

TOKAMAK PLASMAS

Candidate:

Andrea Casolari

Supervisors:

Dr. Paolo Buratti

Prof. Francesco Pegoraro

Aknowledgements

Contents

Nomenclature

Chapter 1

Introduction

1.1

Background and motivations

1.2

Purpose of this work

1.3

Outline of the thesis

Chapter 2

Toroidal geometry and neoclassical

theory

2.1

Magnetic field related concepts

2.2

MHD equilibrium

2.3

Particle orbits

2.4

Transport ordering

2.5

Collisionality

2.6

The kinetic equation

2.7

Fluid equations

2.8

The drift-kinetic equation

2.9

Parallel particle flux

2.10

Flow across flux surfaces

2.11

Transport in the banana regime

2.12

Ion transport

2.13

The parallel viscous force