Turbulent sources in axisymmetric plasmas This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 61.175.193.51 This content was downloaded on 03/10/2013 at 08:27 Please note that terms and conditions apply. 2009 Plasma Phys. Control. Fusion 51 075002 (http://iopscience.iop.org/0741-3335/51/7/075002) View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Turbulent sources in axisymmetric plasmas
This content has been downloaded from IOPscience. Please scroll down to see the full text.
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IP Address: 61.175.193.51
This content was downloaded on 03/10/2013 at 08:27
Please note that terms and conditions apply.
2009 Plasma Phys. Control. Fusion 51 075002
(http://iopscience.iop.org/0741-3335/51/7/075002)
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Received 23 October 2008, in final form 24 March 2009Published 13 May 2009Online at stacks.iop.org/PPCF/51/075002
AbstractSuccessful operation of tokamaks and other magnetic confinement schemes offusion interest rely on the tailoring of the parallel momentum/current densityand temperature profiles via resonant absorption of externally injected waves.Similarly, it is to be expected that a turbulent spectrum of waves, internallygenerated to free the energy stored in the gradients of the equilibrium profiles,could transfer locally momentum and energy to the particle degree of freedomof the plasma. Turbulent sources stem out nicely from the action-angle transportformalism, as a detailed derivation of the general transport law from the collisionoperator (which includes both the diffusion and the friction coefficients) inaction-space shows. The special case of magnetic turbulence is considered, andexplicit expressions for the electron parallel momentum and energy sources arepresented. An interesting feature of the sources resides in their dependenceon the first and second powers of the safety factor derivative, a dependencethat is often found in turbulent fluxes as well. One term in the energy sourcedepends, in a determinant way, also on the relative magnitude of the electronand ion temperature. This dependence, an output of the retention of the frictionterm in the collision operator, leads to an energy flow that is always directedfrom the hotter to the cooler species, a desirable property that is missed whena quasilinear approach is employed.
1. Introduction
It is likely that a steady-state fusion reactor based on the tokamak concept will necessitate,continuously or at specific times during its operation, externally injected rf power to heat theplasma and/or drive current. Indeed, methods of rf heating and current drive are routinelystudied in experiments nowadays. The basic principle underlying non-inductive heating andcurrent drive is simple: if a beam of waves with energy hω and longitudinal momentum hk‖injected into a plasma is resonantly absorbed by a plasma species with mass m and densityN at the rate of N quanta per unit volume per unit time, the plasma species acquires themomentum Nmv‖ = hk‖Nτ and the energy Nmv2/2 = hωNτ per unit volume and inthe time interval τ , where v is the particle velocity. On the other hand, in a magnetically
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
|B|~
r
equilibrium
perturbedprofile
profile (e.g. T)
U
U
Γ
2
Figure 1. Flux and source due to turbulence. Turbulence both acts as a (positive or negative)localized source (U ) and induces radial transport ()
confined plasma instabilities free the energy stored in the equilibrium profiles generating aturbulent spectrum containing a certain amount of wave power. Similarly to the just describedexternally induced heating and current drive, this ‘internally injected’ power associated withthe turbulence might contribute in a significant way to the localized heating/cooling as well asto the generation/damping of current density.
In spite of their potentially important impact, turbulent sources have so far received littleattention compared with turbulent fluxes, and are often neglected in numerical computationof plasma transport. This general attitude is founded on the assumption that the sourcecontribution should be small in most turbulent plasmas. In this paper we reconsider the issueof turbulent sources in fusion-relevant plasmas in which collisional effects are secondary withrespect to wave–particle resonant interactions. Our main objective is to derive expressionsfor the turbulent sources which are simple enough to be easily incorporated into the linearmomentum and the energy fluid equations, and to verify their potential to influence the evolutionof plasma discharges. In pursuing this goal, we will not focus on the turbulent spectrumcharacterizing the turbulence; instead we will adopt an ansatz for it. What we are interested inis to find out the general expression of the sources, and in particular their dependence on theequilibrium profiles. Detailed calculations for the turbulent spectrum are outside the scope ofthe paper.
We pursue our goal of studying turbulent sources by employing an elegant and compactkinetic formalism based on action-angles, initially put forward in the fusion research contextby Kaufman [1], to derive both the parallel momentum and the energy source due to magneticturbulence. Action-angle variables are adopted because, since they are adiabatic invariants thatchange only because of resonant wave–particle interactions, they represent the most convenientphase-space variables to describe resonant transport. Whenever a resonance breaks one or moreinvariants of the unperturbed motion of the particles, the latter acquire momentum and energy,and undergo a random walk in action-space. In the theory, these two effects are representedby a source term (U ) and a flux term () appearing in the radial balance of momentum andenergy:
∂χ
∂t= 1
r
∂
∂rr + U,
where χ = mv‖ or χ = mv2/2. These two different effects are schematized in figure 1.Instead of following the quasilinear approach, as in Kaufman’s original work, we follow
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
the ‘self-consistent’ approach, as formulated by Hitchcock et al [2] and Mynick [3], andwhich is the only formulation that gives results valid for steady-state fluctuation spectra. By‘self-consistent operator’ we mean a collision operator that includes, besides the diffusioncoefficient, a friction term which physically represents the polarization drag felt by the scatteredparticles1. Action-angle transport has been employed to study several issues in axisymmetricplasma transport, both following the quasilinear [4] and the self-consistent [5] approach. Inthe existing literature, to our knowledge, the only work carried out within the action-angleframework and presenting detailed expressions of the momentum and energy sources (as well asthe fluxes) due to a supra-thermal level of low-frequency fluctuations is the one of Mahajan et al[6]. Since the latter work adopts the quasilinear formulation, our paper may be considered as anextension of it to include the effect due to the back-reaction of the particles on the fluctuations.Our main goal is to find the dependence of the sources on the equilibrium plasma profiles, toidentify the relevant transport coefficients and to quantify the potential impact that turbulentsources can have on the plasma evolution. The results are specialized to describe the transport ofpassing electrons induced by magnetic turbulence in low-collisionality axisymmetric plasmas,so that the main underlying physical mechanism is the resonant interaction between waves andparticles.
An interesting outcome of our analysis is the dependence of the turbulent sources on thesafety factor profile, a dependence that originates from the particle drift in inhomogeneousgeometries. At locations where the safety factor is flat, the sources are zero (to lowest order).Wherever the gradient of the safety factor is non-zero, it regulates both the magnitude of thesources and the sign of some of their contributions. Given the variety of safety factor profilesthat characterize tokamak discharges, it is this dependence that could give relevance to thesources. Note that a dependence on the safety factor profile is not new to turbulent transport.For example, the assumption of turbulent equipartition [7] leads to particle fluxes that areproportional to the radial derivative of the safety factor [8, 9]. This is the first time, to ourknowledge, that this dependence is shown to exist, and is presented in detail, in the sourceterms as well.
The rest of the paper is organized as follows. In the next section, we review the fundamentalideas underlying the self-consistent transport theory in action-angle variables, and introducethe specializations required to deal with toroidal axisymmetric geometries (in particular,tokamaks). A detailed derivation of the general radial transport law is presented in section 3,making transparent the origin of the source terms as opposed to the fluxes. The specializationon the parallel momentum and on the energy transport equations is also presented in this section.In section 4, we discuss the fluctuation spectrum and the pseudo-thermal approximation whichdescribes magnetic micro-turbulence. The procedure to evaluate the momentum and the energysource for passing electrons is presented in section 5, and a numerical investigation of someproperties of the sources is carried out in the following section. We summarize and discussthe results in section 7.
2. Action-angle transport theory in toroidal geometry
We consider an axisymmetric toroidal device which confines a plasma by a slowly varyingmagnetic field. In this context, the collision operator can be expressed in a particularly
1 The adjective ‘self-consistent’ is often referred to a theory in which the fluctuation spectrum is evaluated by couplingthe moment equations with the appropriate field equations. We stress that we do not do that in our work; as describedin section 4, we simply adopt an ansatz to model the spectrum. By the term ‘self-consistent operator’ we only meanthat the operator includes both diffusion and friction terms, as opposed to the quasilinear operator which retains onlythe diffusion term, and thus misses the back-reaction of the particles on the fields.
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
simple and general form [1–3] by (i) introducing action-angle variables to describe theparticle motion and (ii) expressing the fields as a sum over plasma normal modes. Whilethe use of action-angle variables leads to a natural inclusion of toroidal effects (such asparticle trapping, drifts, etc), the normal mode expansion permits, through a formal solutionof Maxwell’s equations, the inclusion in the transport coefficients of fluctuation spectrathat are valid in fully inhomogeneous geometries. In the present section we summarize(following closely the presentation of [10]) the main ingredients of the action-angle approachto transport, referring the reader to [4, 5] (and references therein) for a more comprehensivedescription of the theory. In particular, in section 2.1 we define the actions and the modedecomposition, in section 2.2 we present the self-consistent version of the operator (oftenreferred to, in the literature, as the ‘generalized Balescu–Lenard’ (gBL) operator) and insection 2.3 we specialize on the case of large aspect-ratio tokamaks with a strong toroidalfield.
2.1. Action-angle variables and normal mode decomposition
To specify the action-angle variables we adopt, together with the usual Cartesian coordinateset x, the flux coordinates ξ = (α, θ, ζ ), where α is a minor radius-like variable constanton a flux surface, and where θ and ζ are, respectively, the poloidal and the toroidal angle,both of periodicity 2π . To be specific, we will take α to be the toroidal flux functionα ≡ t/(2π), where t is the toroidal flux enclosed by the flux surface. (From now on,the subscript t(p) stands for toroidal (poloidal).) To express some of the results in a moreperspicuous way we also introduce an ‘equivalent’ cylindrical radial coordinate r , defined bythe relation t(α) = 2πα = ∫
dSB · ζ ≡ πr2B0,t (an over-hat indicates unit vectors), wherethe reference field B0,t is chosen to be the toroidal magnetic field on the magnetic axis. Thus,r = r(α) = [2α/B0,t]1/2. The actions parametrizing the phase-space particle point are taken tobe [1, 4] J = (Jg, Jb, Jζ ), Jg = µ0M
2c/q being the gyro-action (where µ0 ≡ v2⊥/(2B0) is the
(lowest order) magnetic moment, B0 the total equilibrium magnetic field, v⊥ the component ofthe particle velocity perpendicular to the magnetic field, M and q the particle mass and charge,respectively and c the speed of light), Jζ the toroidal angular momentum,
Jζ = MR2ζ − q
cψp(α) (1)
(where ψp ≡ p/2π is the poloidal flux function and R the major radius) and Jb beingthe longitudinal invariant or bounce action, equal to the toroidal flux enclosed by a driftorbit. Denoting the projection on the poloidal cross-section of the guiding center trajectorywith α = α(θ), the bounce action can be written as Jb = ∮
(dθ/2π)(q/c)α(θ; H0, Jg, Jζ ),where the triplet (H0, Jg, Jζ ) singles out one particular particle trajectory. The conjugateangles Θ = (g, b, ζ ), which are cyclic coordinates (i.e. ∂H0/∂Θ = 0 where H0 is theunperturbed Hamiltonian), represent respectively the orbit-averaged gyro-phase, the phaseof the bounce motion and the bounce-averaged toroidal angle. As usual in Hamiltoniantheory, they are obtained by derivatives of the appropriate generating function with respectto the conjugate new action. Because J is constant in the absence of perturbations, theunperturbed motion is trivial: Θ develops linearly in time. The corresponding set of bounce-averaged gyration frequency, bounce frequency and bounce-averaged toroidal (or banana)drift are defined in terms of the unperturbed Hamiltonian H0, which is a function of J only:Ω ≡ Θ = ∂H0(J)/∂J.
To linearize, the electrostatic and the magnetic potentials are decomposed into unperturbedand perturbed parts, = 0 + 1, A = A0 + A1. Because J is a constant of the unperturbedmotion and Θ is a cyclic coordinate, we write H(J,Θ) = H0(J) + h(J,Θ) + · · ·. The
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
unperturbed Hamiltonian, H0 = (1/2M)[p − (q/c)A0]2 + q 0 = Mv2/2 + q 0 (where pis the canonical angular momentum, p = Mv + qA0/c, and v, v2 is the particle velocity)is allowed to change, due to variation of the background fields, but only very slowly (quasi-statically) on a transit time scale. The first order perturbing Hamiltonian h = q 1−(q/c)v ·A1
is expanded in a Fourier series in the ignorable and periodic angle coordinates, h(J,Θ; t) =∑ h(J, ; t) exp(+i · Θ). The triplet of integers = (g, b, ζ ) singles out each one of
the harmonics of the particle perturbing Hamiltonian, or, analogously, of the orbital motion.Note that in the general case these integers differ from the usual poloidal (ma) and toroidal(na) mode numbers entering the Fourier decomposition of a field with respect to θ and ζ ,and indicating its spatial dependence. The space-time Fourier transform of the perturbingHamiltonian, h(J, ; ω) = ∮
The gBL transport theory [5] extends the validity of Kaufman’s original quasilinear action-angle theory [1] in two ways, i.e. (i) is valid for steady-state fluctuation spectra and (ii) isapplicable, although through a simplifying approximation (the ‘pseudo-thermal ansatz’ of [5]),to realistic plasma turbulence. The latter point will be discussed in section 4. The formerextension is achieved by the inclusion in the collision operator of the polarization drag felt bythe scattered particle, which represents the back-reaction of the particles on the fields. Thedistribution function of the scattered species 1 evolves according to [3]
where F is the friction vector that considers the polarization field. Indicating with 2 thescattering species and witha each of the normal modes constituting the fluctuation spectrum, wewrite [5] D(J1) = ∑
2
∑1,2
11D0(J1, 1, 2) and F(J1) = ∑2
∑1,2
12 · F0(J1, 1, 2),where[
D0(J1; 1, 2)
F0(J1; 1, 2)
]=
(2π
M2
)3 ∫dJ2Q(1, J1; 2, J2)
[f0(J2)
∂J2f0(J2)
], (3)
Q(1, J1, 2, J2) ≡ 2πδ(1 · Ω1 − 2 · Ω2)∑
a
|Ca(1, J1, 2, J2, ω = 2 · 2)|2 , (4)
and Ca(1, J1, 2, J2, ω) ≡ [4πha(1, J1, ω)h∗a(2, J2, ω)]/Naa(ω). In the latter expression,
a(ω) is the eigenvalue of the Maxwell operator, and Na is a normalization factor. Equation (2)describes the slow (compared with the characteristic particle frequencies) evolution of f0(J1; t)
as a result of a random walk in action-space, induced by the normal modes a emitted byspecies 2. We retain a summation over 2 to consider the case of more than one scatteringspecies (ions, impurities, etc). The coefficient Ca , the ‘coupling coefficient’, measures theeffectiveness of mode a in coupling particles 1 and 2. The Hamiltonian ha = q a
1 −(q/c)v·Aa1
to be used in Ca must be calculated using the expression of the particle velocity valid in themagnetic geometry of interest, toroidal in our case, and evaluating the fields 1 and A1 alongthe particle orbit.
It can be shown that the gBL operator is characterized by the following three properties[3, 5]: (i) interaction between particles of the same species does not produce any net particletransport; (ii) the particle fluxes of the two species are equal; (iii) the transport is independentof the radial electrostatic potential. Note that the presence of the friction vector F is essential
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
for the operator to possess these properties. The self-consistent approach has already beenemployed to study various aspects of particle and energy transport driven by thermodynamicforces [5, 10, 11].
2.3. Specialization to large aspect-ratio tokamaks
Following tokamak theory, we adopt the large aspect-ratio and the small gyro-radius orderings,ε ≡ |(R − R0)/R0| ∼ |r(α)/R0| , δ ≡ ρ/L 1, where ρ ≡ vth/g is the gyro-radius, L is alength characterizing the variation of the equilibrium quantities, vth = (2T/M)1/2 is the thermalvelocity (T being the temperature in energy units) and R0 is the major radius. In these orderings,the safety factor is well approximated by its cylindrical version, qsaf rB0,t/(R0Bp). Since thetoroidal field is predominant (Bp/Bt ≈ ε), we can approximate the modulus of the unperturbedmagnetic field as
B ≈ B0,t
[1 − r(α)
R0cos θ
]= B0,t
[1 − r(α)
R0
]+ B0,t
r(α)
R02 sin2 θ
2. (5)
To the lowest order in the gyro-radius, particles in a large aspect-ratio tokamak simply streamalong the field lines with a parallel velocity that can be expressed in terms of the Hamiltonian:v‖ ≡ σu = [(2/M)(H0 − q 0 − Mv2
⊥/2)]1/2, where σ = ±1 (indicating the two possibledirections of motion along the field lines) and u is the parallel flow speed. In general, becauseof the twisting of the magnetic field lines, the toroidal velocity differs from the parallel velocity.In view of the tokamak ordering B0 B0,t , however, we approximately set vt ≈ v‖, and fromequation (1) we obtain
Jζ = MRvt − q
cψp(α) ≈ MRσu − q
cψp(α). (6)
Using expression (5) for the unperturbed magnetic field, we can recast the parallel flowspeed as u(θ; J) = u0(µ0)[κ2(J) − sin2(θ/2)]1/2, where we have defined the quantitiesu0(µ0) ≡ 2 [µ0B0ε]1/2 and κ2(J) ≡ H0(J)−q 0[α(J)]−µ0MB0(1−ε)/(2µ0MB0ε), andwhere any θ dependence of 0 is neglected. The trapped region is identified by κ = [0, 1],while the untrapped region by κ = (1, +∞].
In the expansion of the radial coordinate, α = α0 + α1 + · · ·, the lowest order contributionα0 represents the toroidal flux enclosed by the magnetic surface around which the particlemotion evolves, and α1 the excursion from the surface due to drifts. We can thereforeTaylor expand the unperturbed poloidal flux function as ψp(α) = ψp(α0) + (∂ψp/∂α)α0α1 =ψp(α0) + [1/qsaf(α0)]α1 (where qsaf is the safety factor), obtaining from equation (6):Jζ + (q/c)ψp(α0) + (q/c)[α1/qsaf(α0)] = M(R0 + ε cos θ)v‖. At zero and first order we have
Jζ +q
cψp(α0) = 0, and
q
c
α1
qsaf(α0)= MR0v‖. (7)
α0 depends only on Jζ , α = α0(Jζ ), while α1 depends on both J and θ because u = u(θ; J)
does so. Solving the second relation in equation (7) we find for the α1 correction: α1 =[cqsaf(α0)/q]MR0v‖(θ; J).
3. Radial transport law
Contrary to conventional collision operators in (x, p) space, action-space collision operators arenon-local, i.e. do not conserve particles, momentum and energy at each spatial point. Transportfluxes can thus be obtained directly by simply taking moments [4]. Paralleling the procedurepresented in [4] (in the quasilinear context), in this section we derive the general transport
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
equation, stressing the origin of the flux and the source term. The result is very general, sinceno specializations have been done on the actions and on the fluctuation spectrum yet. The onlyapproximation consists of the adoption of a drifting-Maxwellian ansatz for the lowest orderdistribution function.
Let χ(x, v; t) be some quantity whose mean with respect to the distribution function(suppressing species label for the moment),
χ(x; t) ≡∫
dvχ(x, v; t)f0(x, v; t), (8)
is of physical interest. In particular, the moments χ = 1, Mv‖ and Mv2/2 (where v‖, v are theparallel and total particle speed, respectively) will lead to the density, parallel momentum andenergy transport laws, respectively. We define the flux-surface average of χ as its normalizedvolume average restricted to a selected flux surface α:
〈χ〉α (α, t) ≡ 1
V ′
∫dxδ[α(x, t) − α] χ(x; t), (9)
where V ′ ≡ dV(α)/dα, with V(α) the volume inside the flux surface α. The derivationof the transport law begins by taking the time derivative of (9), after having made useof definition (8) and of the approximation ∂V ′/∂t 0 (stationary toroidal flux surfaces):∂ 〈χ〉α /∂t = (1/V ′)
∫dx
∫dv(∂/∂t) [δ(α − α)f0(x, v, t) χ(x, v, t)], or, in terms of action-
angle variables (M3∫
dx dv = ∫dJ dΘ),
∂
∂t〈χ〉α = 1
V ′ M3
∫dJ
∫dΘ
∂
∂tχ(J,Θ, t)δ[α(J,Θ, t) − α] f0(J,Θ, t) . (10)
The time dependence of α arises from the quasi-static variation of the equilibrium. As an ansatzto accomplish closure, we assume a displaced, locally Maxwellian distribution function, so asto be able to evaluate transport fluxes and sources including the presence of a toroidal current.Keeping only the first order terms in |V‖/vth| 1, we thus consider f0(J, t) = fM(J, t)1 +V‖[α(J, t)]P [α(J, t)]/T [α(J, t)] where V‖ is the parallel drift (flow) speed, P ≡ Mv‖ isthe parallel particle momentum, and fM(J, t) = N(α)M3/2/[π3/2 23/2T (α)3/2] exp−[H0 −q 0(α)]/T (α). K0 ≡ H0−q 0 = Mv2/2 is the (unperturbed) kinetic energy when a particleis at α = α0. Note that in general α = α(J,Θ; t); here however, f0(J) is the Θ-averaged lowestorder solution, so that α = α(J; t). Since for far-untrapped particles the quantity α1 is small,we evaluate all the quantities (such as N , T ) at α = α0(Jζ ). Under this Maxwellian ansatz,the derivation of the transport law proceeds as follows. Using the chain rule of differentiationand the antisymmetry property of the delta function, we have, from equation (10),
∂
∂t〈χ〉α = 1
V ′ M3
∫dJ
∫dΘδ[α(J,Θ, t) − α]f0(J, t)
∂χ(J,Θ, t)
∂t
− ∂
∂α
∫dJ
∫dΘδ[α(J,Θ, t) − α]f0(J, t)χ(J,Θ, t)
∂α(J,Θ, t)
∂t
+∫
dJ∫
dΘδ[α(J,Θ, t) − α]χ(J,Θ, t)∂f0(J, t)
∂t
, (11)
where all time derivatives are performed at constant (J,Θ). The first two terms on the right-hand side (RHS) can be recast as⟨
∂χ
∂t
⟩α
and − 1
V ′∂
∂αV ′
⟨χ
∂α
∂t
⟩α
. (12)
The second term in equation (12) is a generalized, collisionless version of the trapped-particleWare–Galeev pinch, as shown in [4]. We will present explicit expressions of this term in
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
section 3.1. The last term inside the curly brackets in equation (11) is what gives rise to fluxand source terms. Let us denote this term with T . Introducing in it the kinetic equation (2) aswell as the definitions of D and F we obtain, after having performed an integration by parts,
T = −∑1,2
∫dJ1
∫dΘ1D0(J1,1,2)1 ·
∂f0(J1;t)∂J1
1 · ∂
∂J1δ[α1(J1,1;t)−α]χ1(J1,Θ1;t)
+∑1,2
∫dJ1
∫dΘ12 ·F0(J1,1,2)f0(J1;t)1 ·
∂
∂J1δ[α1(J1,1;t)−α]χ1(J1,Θ1;t).
The action-derivative acts now on a product of two terms, and will thus leadto two contributions to T . Considering that ∂δ[α1(J1, 1; t) − α]/∂J1 =−(∂/∂α) (∂α/∂J1)δ[α1(J1, 1; t) − α], we see that the first of these two contributions(proportional to ∂α/∂J) will be a flux, while the second term (proportional to ∂χ/∂J) willbe a source. Introducing the two factors
Xχ(J, t, ; α) ≡∫
dΘ(2π)3
χ(J,Θ, t) · ∂α(J,Θ, t)
∂Jδ[α(J,Θ, t) − α], (13)
Yχ(J, t, ; α) ≡∫
dΘ(2π)3
· ∂χ(J,Θ, t)
∂Jδ[α(J,Θ, t) − α], (14)
we obtain
T = (2π)3∑1,2
∂
∂α
∫dJ1D0(J1, 1, 2)1 ·
∂f0(J1; t)
∂J1Xχ(J1, 1; α, t)
− (2π)3∑1,2
∫dJ1D0(J1, 1, 2)1 ·
∂f0(J1; t)
∂J1Yχ(J1, 1; α, t)
− (2π)3∑1,2
∂
∂α
∫dJ12 · F0(J1, 1, 2)f0(J1; t)Xχ(J1, 1; α, t)
+ (2π)3∑1,2
∫dJ12 · F0(J1, 1, 2)f0(J1; t)Y χ (J1, 1; α, t) .
Both the diffusion and the drag term in equation (2) have thus led to a flux contribution(proportional to the factor Xχ ) and a source contribution (proportional to Yχ ) to the transportlaw for χ . Using these results, as well as equation (12), the latter law takes the form
∂
∂t
⟨χ1
⟩α
−⟨
∂χ1
∂t
⟩α
+1
V ′∂
∂αV ′
⟨χ1
∂α(t)
∂t
⟩α
+1
V ′∂
∂αV ′1(α) = U1(α), (15)
where the flux 1 and the source U1 are given by[1(α)
U1(α)
]=
∑2
[12(α)
U12(α)
]= −
∑2
∑1,2
(2π
M1
)3 1
V ′
∫dJ1 [D0(J1; 1, 2)
×1 · ∂f (J1; t)
∂J1− 2 · F0(J1; 1, 2)f (J1; t)
] [Xχ1(J1, 1; α, t)
Y χ1(J1, 1; α, t)
]. (16)
Looking at equation (15), we first note that, together with the usual time variation of the flux-average of the quantity of interest χ , the first term on the left-hand side (LHS), we have a term,the second term, given by the flux-average of the time derivative of the quantity of interest χ .Physically this second term is needed to take into account possible deformations of the flux
8
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
surface in time. The third term derives from the time derivative of the delta function whichlocalizes the flux surface. Therefore it takes care of the fact that the magnetic flux enclosedby the flux surface varies as a result of the slow variation of the equilibrium fields. As we willsee better later on, this term is a generalized version of the Ware–Galeev pinch. The fourthterm is the flux term, and the fifth term (on the RHS) is a source term. We observe that inthe action-space approach, radial transport occurs even in the strictly Maxwellian case, sincediffusion in action-space entails diffusion in coordinate space [4]. This is so because f (J)
is an exact solution of the unperturbed Liouville equation, and thus contains, in addition tothe usual local Maxwellian used in more standard approaches to transport, the collisionlesscorrection (in banana width to minor radius) that leads to radial transport. To proceed, we needto calculate the J derivative of f , as required in equation (16). First we note that the covariantcomponent of the toroidal canonical momentum is Jζ ≡ pζ = p · ζ = [Mv + (q/c)A0] · ζ =Mvζ + (q/c)A0,ζ = MRvt − (q/c)ψp (where vt = Rvζ is the toroidal velocity), and thereforewe have P ≡ Mv‖ ≈ Mvt = (1/R)Jζ + (q/c) ψp[α(J)]. We obtain
· ∂
∂Jf0(J; t) = fM
−
(1 +
V‖PT
) · ΩT
+V‖T
G
+ g
[AN +
K0
TAT +
V‖PT
(AN +
K0
TAT + AV
)], (17)
where we have introduced the factors
g ≡ · ∇Jα, G ≡ · ∇JP, K0 ≡ H0 − q 0 = Mv2
2, (18)
as well as the thermodynamic forces (a prime indicating a derivative with respect to α)
AN = N ′
N+
q
T ′
0 − 3
2
T ′
T, AT = T ′
T, AV = V ′
‖V‖
− T ′
T. (19)
Using equation (3), we arrive at the following final expressions for the total flux and source:[1(α)
U1(α)
]=
∑2
∑1,2
1
V ′
(2π
M1
)2 ∫dJ1
∫d6z2Q(1, J1; 2, J2)
× fM(J1)fM(J2)
[Xχ(J1, 1; α, t)
Y χ(J1, 1; α, t)
]A(J1, J2, 1, 2), (20)
where to shorten the notation we have defined z ≡ (J,Θ) and
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
Note that the Θ integrations are present only in the X or Y factors, the remaining termsnot depending on the angles. equation (15) with definitions (18)–(21) is the radial transportlaw which describes the time-evolution of any flux-surface averaged moment of the scatteredparticle distribution function. Note that in order to evaluate equation (20) we need to specifythe term · Ω and the factors X, Y , g and G defined in equations (13), (14) and (18). Thegeneral expressions of these factors have been reported in [12]. Here we will present only theirlimiting expressions valid in the far-untrapped case.
3.1. Momentum and energy transport equations
Considering χ = Mv‖ Mvζ and expressing the results in terms of r using 〈· · ·〉α =[1/(rB0,t)]〈· · ·〉r , ∂/∂α = [1/(rB0,t)]∂/∂r , equation (15) leads to the parallel momentumtransport equation:
Va
∂⟨MN(x; t)V‖(x; t)
⟩r
∂t− Va〈q1NEt〉r +
∂
∂r〈VaVfMNV‖〉r S(1 − κ)
+∂
∂rVa
V (r) = (rB0,t)VaUV , (22)
where Va = 4π2R0wr is the volume of the toroidal shell (centered at r and of width w) overwhich the mode a is non-zero, and the flux and source terms are again given by (20). In thesecond term on the LHS, Et is the toroidal induction electric field Et = (1/cR)(∂ψp/∂t),where the time derivative is evaluated at a fixed position in space. While the poloidal magneticfield can remain constant (if the external fields remain so), the flux surface on which ψp hasa particular value will move. In a tokamak, due to the smallness of Bp, the velocity of theflux surfaces is very large. As suggested by the presence of the velocity of a flux surface,Vf ≡ cEt/Bp, the last term in the first line of equation (22) is a collisionless version of theWare–Galeev pinch, effective only for trapped particles (and this is the reason for the stepfunction S(1 − κ) yielding 1 (0) for κ < 1 (κ > 1)). The source term UV accounts for theparallel momentum generation/damping by the fluctuations.
Considering the χ = Mv2/2 = H0 − q 0 moment, we obtain the energy transportequation:
Va
∂⟨
32N(x; t)T (x; t)
⟩r
∂t− Va〈q1NV‖Et〉r +
∂
∂r
⟨VaVf
3
2NT
⟩r
S(1 − κ)
+∂
∂rVa
T (r) = (rB0,t)VaUT , (23)
where the fluxes and source expressions are the usual ones with Xχ=M1v21/2 and Yχ=M1v
21/2.
On the basis of what has already been said in relation to the momentum balance, all terms in(23) have an easy interpretation. The source term, in particular, describes the turbulent energyexchange between the plasma and the waves, due to the work done by the total electric field:〈E · j〉 (wave heating).
4. Fluctuation spectrum for magnetic micro-turbulence
As for the standard Balescu–Lenard operator, the spectrum contained in D and F is of the‘thermal’ type, and does not correctly model realistic turbulence [3]. This limitation has beenovercome in [5], although in an approximate way, with the adoption of a supra-thermal spectrum(called the ‘pseudo-thermal’ spectrum), which retains the structure of the original thermalspectrum (therefore maintaining the required properties of the collision operator) but replaces
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
its form so as to better represent experimental features. Here we only outline the essence of thisapproximation, since it is well documented in the literature. It is assumed that the eigenvalue a
is non-linearly modified from its thermal value in such a way that the turbulent vector potentialdriven by species 2 has the form |Aa|2 ∝ B2(2) exp−k2
⊥/[2(k⊥)2]−k2‖/[2(k‖)2]/(k⊥)2,
where k⊥ ∼ ρ−1gi and k‖ ∼ L−1
s (with Ls ∼ qsafR0 the shear length) are the spectral widthssatisfying k⊥ k‖. The overall strength of the magnetic fluctuations induced by species2, indicated with B2(2) ≡ 〈B2
r 〉, is assumed to be non-zero and approximately constant onlywithin the volume of the narrow toroidal shell Va = 4π2rawaR0. The ‘pseudo-thermal,magnetic micro-turbulence’ version of equation (20) is[
12(α)
U12(α)
]=
∑ra
∑k
∑1,2
1
V ′(r)
(2π
M1
)3 ∫Va
dJ1 fM(J1)
(2π
M2
)3 ∫Va
dJ2fM(J2)
×[Xχ1(J1, 1)
Y χ1(J1, 1)
]A(J1, J2, 1, 2) 2πδ[1 · Ω1(J1) − 2 · Ω2(J2)]
×|Ca|2δ(ζ,1 − na) δ(ζ,2 − na)J2g1
(zg1) J 2b1−ma
(zb1)J2g2
(zg2) J 2b2−ma
(zb2),
(24)
where |Ca|2 = (q1/c)2|Aa
‖|2|v‖1|2|v‖2|2/(VaN2〈|v‖2|2〉), 〈· · ·〉 indicates a thermal average,q is the particle charge and A is defined in equation (21). Here, zg = k⊥ρg where ρg is thegyration radius and k⊥ the perpendicular wavevector, and zb = [(krrd)
2+(mθd+nζd)2]1/2 where
(rd, θd, ζd) are the amplitudes measuring the extent of the particle excursion from the field linesin the course of a transit period and (kr, m, n) are the radial wavevector, the poloidal and thetoroidal modenumber. The gyro and bounce-related Bessel functions quantify the modificationof the field–particle coupling due to drifts and the modulation of v‖ by the magnetic well [13]. Inthe following we assume ωa < g for both species, and therefore neglect effects from all gyro-harmonics except g = 0. It should be noted, however, that the pseudo-thermal ansatz limitsthe validity of the operator, and thus of equation (24), to strictly stable plasmas. A turbulentgeneralization of the operator would be required to remove this limitation, and obtain a trulyself-consistent transport theory of strongly turbulent, inhomogeneous plasmas [14].
5. Source evaluation
We evaluate the source U12 defined in equation (24) for the case of far-untrapped electrons.As already pointed out, in magnetic turbulence ωa g1,2 and therefore we set from theoutset gj = 0, for j = 1, 2. As a consequence the gyro-related summations drop out andthe contribution of the gyro-related Bessel functions reduces to J 2
0 (zg1)J20 (zg2). In the far-
untrapped limit the Yχ
1 factor reduces to[Y1(χ = M1v‖,1)Y1(χ = M1v
21/2)
]
[1/v‖,1
1
] [(1 · Ω1)∞ + ζ,1cqsaf
∂ 0
∂α0
]
=[
1v‖,1
]1
R0
(b1
qsaf+ ζ1
)+
[v‖,1v2
‖,1
]ζ,1
M1c
q1
∂qsaf
∂α0,
(25)
where the notation (· · ·)∞ means that the term inside the parentheses needs to be evaluatedin the far-untrapped limit κ → ∞. The following explicit expression for (1 · Ω1)∞ hasbeen used to obtain the second form of Y
χ
1 : ( · Ω)∞ = b[v‖/(R0qsaf)] + (v‖/R0)ζF
where F ≡ 1 + (Mc/q)R0v‖(∂qsaf/∂α0) − cqsaf(R0/v‖)(∂ 0/∂α0). In the same limit and
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
approximation, the expressions for the factors g and G defined in equation (18) (and appearingin the driving term A) are, respectively,
g (
cqsaf
q
) [R0
v‖( · Ω)∞ − ζF
]= cb
q(26)
and
G [
1/v‖1
] [( · Ω)∞ + ζ cqsaf
∂ 0
∂α0
]
=[
1v‖
]1
R0
(b
qsaf+ ζ
)+
[v‖v2
‖
]ζ
Mc
q
∂qsaf
∂α0. (27)
There are three steps that need to be performed in order to evaluate the source: sum in-space, integration in J-space and sum in k-space. Introducing in equation (24) the limitingexpressions for the factors Yχ , g and G, as well as V(r)′ = (2π)2R0r , and the followingapproximation of the action integral∫
Va
dJj M2j R0
B0
∫dαj
∫ +∞
−∞dv‖j
∫ +∞
Mj v2‖j /2
dK0j ,
which is valid in the far-untrapped limit [6], we obtain
U12(α) =∑ra
(2π)3/2N1(M1M2)
1/2
(T1T2)3/2
B0t
rVa
b2(2)
〈|v‖2|2〉 S(k)Ov‖,K0J20 (zg1)J
20 (zg2)
×∑
b1,b1
J 2b1−ma
(zb1) J 2b2−ma
(zb2)δ(1 · Ω1 − 2 · Ω2)q2
1
c2
[1/v1
1
] [(1 · Ω1)∞ + nacqsaf
∂ 0
∂α0
]
×(1)(2)
[(1 · Ω1)∞
T1− (2 · Ω2)∞
T2
]
−V‖1
T1
(2)
v‖1
[(1 · Ω1)∞ + nacqsaf
∂ 0
∂α0
]+
V‖2
T2
(1)
v‖2
[(2 · Ω2)∞ + nacqsaf
∂ 0
∂α0
]
−cqsaf
q1
[R0
v‖1(1 · Ω1)∞ − naF (1)
] [AN1(1)(2) +
M1V‖1
T1AV 1(2)v‖1 + AT 1(1)(2)
K0,1
T1
]
+cqsaf
q2
[R0
v‖2(2 · Ω2)∞ − naF (2)
] [AN2(1)(2) +
M2V‖2
T2AV 2(1)v‖2 + AT 2(1)(2)
K0,2
T2
],
(28)
where we have defined (j) ≡ (1 + V‖,jMjv‖j /Tj ) for j = 1, 2, and introduced the operatorsS(k) ≡ ∑
k exp−k2
⊥/[2(k⊥)2] − k2‖/[2(k‖)2]
/[(k⊥)4k‖] and
Ov‖,K0 ≡∫ +∞
−∞dv‖1|v‖1|2
∫ +∞
M1v2‖1/2
dK0,1 exp[−K0,1/T1]
×∫ +∞
−∞dv‖2|v‖2|2
∫ +∞
M2v2‖2/2
dK0,2 exp[−K0,2/T2], (29)
which act on the factors located to their right. In writing equation (28), we have alreadyperformed the summations over ζ1 and ζ2 by using the delta functions δ(ζ,1 − na) andδ(ζ,2 − na). The dα1 integration can be performed immediately with the help of δ(α1 − α),so that all the quantities of species 1 are evaluated at α = α or, in terms of the equivalentcylindrical radius, at r = r = (2α/B0,t)
1/2. To perform the dα2 integration, we simply assume
12
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
that the equilibrium quantities are constant inside Va , and bring them outside the integral sign.For example, for the generic function f (α), we set∫
dα2f (α2) = B0,t
∫ ra+wa/2
ra−wa/2dr rf (r) B0,tVa
(2π)2R0f (ra). (30)
We observe that the integrations that are determinant in selecting the driving terms in thesources are the velocity and energy integrations. The spatial integrations are not as crucial,and therefore the approximation in equation (30) is acceptable for our goals.
To perform the b summations we transform the sums into integrations over ωb,j ≡b,jb,j with the setting
∑+∞b1,2=−∞ → ∫ +∞
−∞ db1,2 = (1/b1,2)∫
dωb1,2, a step justified bythe presence of resonance-broadening effects. The ωb1 integral can be performed immediatelywith the aid of the remaining delta-function imposing self-consistency. After having performedthe ωb1 integration, we are left with
U12(α)=∑ra
(2π)3/2N1(M1M2)
1/2
(T1T2)3/2
B0t
rVa
b2(2)
〈|v‖2|2〉J20 (zg1) J 2
0 (zg2)q2
1
c2S(k) Ov‖,K0
[1/v1
1
]1
b1b2
×∫
dωb2J2(zb1)(ωb2+naζ2−naζ1)/b1−ma
J 2(zb2)ωb2/b1−ma
[(ωb2 +naζ2
)∞ +nacqsaf
∂ 0
∂α0
]
×(1)(2)(ωb2 +naζ2)
T2 −T1
T1T2−
(ωb2 +naζ2 +nacqsaf
∂ 0
∂α0
)[V‖1
T1
(2)
v‖1− V‖2
T2
(1)
v‖2
]
− cqsaf
q1
[R0
v‖1(ωb2 +naζ2)∞−naF (1)
][AN1(1)(2)+
M1V‖1
T1AV 1(2)v‖1 +AT 1(1)(2)
K0,1
T1
]
+cqsaf
q2
[R0
v‖2(ωb2 +naζ2)∞−naF (2)
][AN2(1)(2)+
M2V‖2
T2AV 2(1)v‖2 +AT 2(1)(2)
K0,2
T2
].
As in [5], we approximate the bounce-related Bessel functions by the pairwise average of theirasymptotic forms: J 2
b(zb) ≡ (1/2)
[J 2
b(zb) + J 2
b+1(zb)] = s(ωb/2, ωb)/(ωb/b), where
ωb ≡ πzbb (the step-like localizing function s(x, y) is defined by s(x, y) = 1(0) forx |y| (x < |y|)). Defining ω′
b2 ≡ ωb2 − mab2 and x ≡ ma(b1 − b2) + na(ζ1 − ζ2),the source can be rewritten as
U12(α) =∑ra
(2π)3/2N1(M1M2)
1/2
(T1T2)3/2
B0t
rVa
b2(2)
〈|v‖2|2〉J20 (zg1) J 2
0 (zg2)q2
1
c2S(k) Ov‖,K0
[1/v1
1
]
×Ib
[(ω′
b2 + mab2 + naζ2)∞ + nacqsaf
∂ 0
∂α0
] (1)(2)(ω′
b2 + mab2 + naζ2)T2 − T1
T1T2
−(
ω′b2 + mab2 + naζ2 + nacqsaf
∂ 0
∂α0
) [V‖1
T1
(2)
v‖1− V‖2
T2
(1)
v‖2
]
−cqsaf
q1
[R0
v‖1(ω′
b2 + mab2 + naζ2)∞ − naF (1)
]
×[AN1(1)(2) +
M1V‖1
T1AV 1(2)v‖1 + AT 1(1)(2)
K0,1
T1
]
+cqsaf
q2
[R0
v‖2(ω′
b2 + mab2 + naζ2)∞ − naF (2)
]
×[AN2(1)(2) +
M2V‖2
T2AV 2(1)v‖2 + AT 2(1)(2)
K0,2
T2
], (31)
where we have introduced the operator Ib ≡ (1/ωb1ωb2)∫
dω′b2s(ωb2/2, ω′
b2)
s(ωb1/2, ω′b2 − x). When expanding the terms inside the parentheses, we see that the
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
operator Ib will act either on 1, ω′b2 or (ω′
b2)2. In the simplifying assumption of small x,
the integral Ib acting on 1 gives [5] Ib 1/ωb,M where ωb,M ≡ max(ωb1, ωb2) (inthe followings, the subscript M (m) to any quantity a means aM(m) ≡ max(min)(a1, a2)). Inthe same limit, we find Ibω
′b2 0 and Ib(ω
′b2)
2 (1/12)(ωb,m)2/ωb,M. For larger valuesof x, the integrals can also be performed analytically, as discussed in [5], leading however toa final expression for the source that is, although more accurate, more complicated and thusless transparent in its physical meaning. To perform the integration in action-space, we firstsubstitute in equation (31) the limiting expressions
(mab2 + naζ2)∞ = 2 · Ω2|b2=ma v‖2k‖ + na
(v2
‖2M2c
q2
∂qsaf
∂α0− cqsaf
∂ 0
∂α0
),
R0
v‖1(mab2 + naζ2)∞ − naF (1) R0
v‖1k‖v‖2 + nac
∂qsaf
∂α0R0
(M2
q2
v2‖2
v‖1− M1
q1v‖1
)− na
R0
v‖2(mab2 + naζ2)∞ − naF (2) R0k‖ − na,
where also k‖ is relative to mode a. The integrations contained in the operator Ov‖,K0
are then trivially performed, many of them being zero because the integrand is odd in theparallel velocity. As a last step we perform the k-sums. There are three different kindsof sums that can be collected: S(k)nak‖, S(k)n2
a and S(k)nama . For magnetic turbulencewith k‖ k⊥ we can approximate na ≡ R0kt = R0(k‖bt − kqbp) −R0kqbp and
ma ≡ rakp = ra(k‖bp + kqbt) rakqbt , where bt ≡ B0,t/B0, bp ≡ B0,p/B0 and q = b × r.The wavevectors kq and k⊥ are related by k⊥ = kr r + kq q. To keep it general, weintroduce a free parameter p which fixes their ratio: p ≡ kq/k⊥. After having convertedthe summations into integrations according to
∑k = [Va/(2π)2]
∫ +∞0 k⊥ dk⊥
∫ +∞−∞ dk‖, we
obtain S(k)mana −[Va/(21/2π3/2)]p2raR0btbp and S(k)n2a [Va/(21/2π3/2)]p2R2
0b2p.
Before presenting the final expression for the sources, it is opportune to order their variouscontributions. A convenient ordering parameter is the inverse scale length of the equilibriumquantities (density, temperature and safety factor) normalized to the electron (poloidal) gyro-radius: ε ≡ ρe,pA 1, where ρe,p = vth,e/e,p, e,p = eBp/(cme) and where A can beAN,j , AV,j , AT ,j (with j = e, i) or Asaf ≡ (dqsaf/dr)/qsaf . For concreteness, to set the valuesof the various plasma parameters we refer to the Tore Supra discharges described in [15].In particular, we take the minor radius a = 0.71 m and the major radius R0 = 2.4 m, thecentral and the edge safety factors equal to q0 = 1.8 and qedge = 9, the central electron andion temperatures equal to Te(0) = 4.8 keV and Ti(0) = 2.5 keV, the central density equal ton(0) = 2 × 1019 cm−3, and the toroidal magnetic field equal to Bt = 3.8 T. These values giveε ∼ ρe,p/LN 1.4 × 10−3, (vth,i/vth,e)
2 3 × 10−4, (k‖/k⊥)2 (ρi/Ls)2 ∼ 3 × 10−8,
(ωb/(k⊥vth,e))2 ∼ 5 × 10−11 and 1/b2
p 6.4 × 102. These results are well approximatedby the following ordering in terms of ε ≡ 10−3:(
vth,i
vth,e
)2
∼ ε ,
(k‖k⊥
)2
∼ ε3,
(ωb,j /k⊥
vth,e
)2
∼ ε4 ,1
b2p
∼ ε−1. (32)
where in some cases, to avoid irrational powers of ε, we have approximated the exponentswith their closest integers (e.g. 1/b2
p = 1/ε0.93 1/ε).According to this ordering, the O(1) e–i momentum source reads
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
The transport coefficient is defined by L12 = ∑ra
p2πN1b2t DRR(1, 2), with DRR(1, 2) =
vth,1(vth,1/b,M)[v2th,2/〈|v‖2|2〉] b2(2)[J 2
0 (zg,1)J20 (zg,2)/πzb,M]. Using vth,e/b,M qsafR0
and 〈|v‖2|2〉 = v2th,i, we see that DRR(e, i) is a generalized Rechester–Rosenbluth coefficient
[16]. For the O(1) e–i energy source we obtain:
(rB0t)UTei = 15 TeLeiA2
safTi
Te
(Ti
Te− 1
)− 3TeLeiAsaf
Ti
Te
×[AN,e +
5
2AT,e + AN,i +
7
2AT,i +
(Ti
Te− 1
) (e
Ti
d 0
dr
)]. (34)
The electrostatic potential appearing inside the square-brackets in equations (33) and (34)cancels out when we explicit the thermodynamic drives A according to equation (19). We willdiscuss UV
ei and UTei in the next section.
6. Analysis of the turbulent sources
Using the definitions in equation (19), we rewrite the momentum and energy sources in termsof the equilibrium profiles as
(rB0t)UVei = −15MeV‖e
(Lei
Ne
)Ti
Te
(1
qsaf
dqsaf
dr
)2
Ne
−3MeV‖e
(Lei
Ne
)Ti
Te
(1
qsaf
dqsaf
dr
) (dNe
dr+
Ne
Ni
dNi
dr+
2Ne
Ti
dTi
dr+
Ne
V‖e
dV‖e
dr
)(35)
and
(rB0t)UTei = 15Te
(Lei
Ne
)Ti
Te
(Ti
Te− 1
) (1
qsaf
dqsaf
dr
)2
Ne
− 3Te
(Lei
Ne
)Ti
Te
(1
qsaf
dqsaf
dr
) (dNe
dr+
Ne
Ni
dNi
dr+
2Ne
Ti
dTi
dr+
Ne
Te
dTe
dr
). (36)
These expressions explicitly show the cancellation of the electrostatic potential.As noted in the section 1, the contribution of the turbulent sources is generally assumed
to be small with respect to the remaining contributions in the momentum and energy balances.The novel result obtained in this paper, i.e. the important dependence on the shape of qsaf ,could give them a potentially more relevant role. To assess the validity of this statement ina proper way, the source terms should be incorporated in a transport code so as to obtaina quantitative estimate of their impact on the plasma evolution. Such a study goes beyondthe scope of this paper. The analysis, presented in this paper, will be limited to a numericalplotting of equations (35) and (36), assuming typical tokamak equilibrium profiles. Despitethe simplicity of our study, the results indicate that at least in some circumstances relevant totokamak operation, turbulent sources could give an important contribution.
As a preliminary step we plot the transport coefficient Lei/Ne. To accomplish that, weneed to specify the electron density and temperature profiles as well as the safety factor profile.In terms of the rescaled radial variable x ≡ r/a, we choose a radial dependence of the kind(1−x2)γ for the first two profiles, assuming the numerical values N(x = 0) = 2×1013 cm−3,Te(x = 0) = 4.8 keV, γN = 1.5 and γT = 2. For the safety factor, we consider the regular(i.e. monotonically increasing with radius) profile qsaf = q0(1 + 0.5x2 + x4), with q0 = 1.1.The numerical value of the minor and of the major radii are taken to be a = 71 cm and
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
(Lei
/Ne)
x 10
4 [cm
2 /s]t
0 0.2 0.4 0.6 0.8 1x [non-dim]
Figure 2. Transport coefficient Lei/Ne for typical tokamak equilibrium profiles and a constantlevel of magnetic turbulence.
R0 = 240 cm, respectively. For the constant level of magnetic perturbation b2 = 1×10−8, theradial dependence of coefficient Lei/Ne is presented in figure 2, providing a section-averagedvalue of 2.4 × 104 cm2 s−1. This is a typical value for an anomalous transport coefficient.
The dependence of the sources on the first or second power of the safety factor profileoriginates from the presence of the term ∂qsaf/∂α0 in expression (25) of the Y factor (differentlyfrom the X factor, entering the flux, that does not have such a term). Such a dependence,anticipated in [10], is ultimately referable to the particles’ drift in an inhomogeneous magneticfield. We note that a dependence on the magnetic shear has been recently found to existin certain components of the turbulent flux, e.g. drift-wave electrostatic turbulence has beenfound to produce an inward flux due to the mechanism of turbulent equipartition [7, 8, 17].Similar contributions to transport fluxes also arise naturally from the same transport theoryadopted in the present work [10, 18]. We therefore choose to analyze the sources by focusingon the effect of different kinds of magnetic-shear profiles, both regular (monotonicallyincreasing toward the plasma edge) and reversed. In particular, we consider the followingmore general model for the safety factor, qsaf = q0(1 + c1x
γ1 + c2xγ2), which reproduces
regular, weakly reversed and strongly reversed magnetic-shear profiles if we assign to the setof coefficients [q0, c1, γ1, c2, γ2] the values of, for example, [1.1, 0.5, 2, 1, 4], [4, −2, 1, 2, 2]and [5.5, −4, 2, 4, 3], respectively. These three profiles are shown in figure 3.
In figure 4 we plot the energy source equation (36) corresponding to the three safety factorprofiles of figure 3. The strongly reversed shear case is denoted with squares, the weakly-reversed shear case with circles and the regular shear case with crosses (the ordinates havebeen arbitrarily normalized, since our only goal was that of investigating the differences in thesource term brought about by different shapes of the safety factor). It is easy to verify thatthe sources go to zero at the location where dqsaf/dr = 0, and reach an extreme at points ofinflexion for the safety factor. The source is always positive only for the regular safety factorprofile. To have a feeling of the actual contribution of the source term, it is convenient tocompare it with the energy flux, the last term on the LHS of equation (23). To this end, weextract T
ei from the total energy flux presented in equation (46) of [10]. For convenience,we rewrite here this flux, together with the source, so as to make evident similarities and
Figure 4. The electron energy source (rB0t)UTei corresponding to the three safety factor profiles
presented in figure 3. The line with crosses refers to the regular case, the line with circles refers tothe weakly sheared case and the line with squares refers to the strongly sheared case. The dashedline at zero is for reference.
differences:
(rB0t)UTei = 3
a2
Lei
NeNeTe
− 1
Lq
[1
LNe
(1 + ηe
)+
1
LNi
(1 + 2ηi
)], (37)
(1
L
)T
ei = 5
2aL
Lei
NeNeTe
−
[1
LNe
(1 + 2ηe
)+
1
LNi
(1 + ηi
)]. (38)
Here, we have neglected for simplicity all terms containing the factor Ti/Te −1, normalized allthe gradient scale lengths L to the minor radius a (an over-hat indicates such a normalization),
17
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
–20
0
20
40
60
80
Ene
rgy
flux
and
sou
rce
[non
-dim
]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8x [non-dim]
Figure 5. The function inside the curly brackets of the energy flux equation (37) and of the energysource equation (38), for the regular safety factor profile (continuous and dashed lines, respectively)and for the strongly reversed safety factor profile (boxes and circles, respectively). The dotted lineat zero is for reference.
introduced the function η = LN/LT and the additional characteristic scale length L ∼ a
to estimate the radial derivative of the flux: ∂Tei/∂r ∼ T
ei/L . It is evident from these twoexpressions that the magnitude of the source and of the flux could be of the same order. Therecould however be a difference in sign, due to the extra factor 1/Lq multiplying the squarebrackets enclosing all the remaining equilibrium profiles of the source. In figure 5 we plotthe functions inside the curly brackets of equations (37) and (38) for the case of regular andstrongly reversed safety factor profile, and for Ti = Te and Ni = Ne. This figure confirms whathas just been been said, showing how the reversed safety factor profile changes the sign of thesource. We conclude our analysis of the energy source by noting that, in the limiting case inwhich the distribution function becomes a true Maxwellian (instead of a local Maxwellian) sothat the gradients are negligible, the only contribution that survives in equation (36) is the onecontaining the factor Ti/Te − 1:
(rB0t)UTei → 15Te
(Lei
Ne
)Ti
Te
(Ti
Te− 1
) (1
qsaf
dqsaf
dr
)2
Ne.
This implies electron-cooling whenever Te > Ti, as it should be. Had we not considered thefriction term (i.e. had we done quasilinear theory), we would not have found that energy flowsfrom the hotter species to the cooler species. This point has first been made in [14].
We have carried out a similar analysis for the momentum source, equation (35), plottingit for the three characteristic safety factor profiles, and for the following model of the toroidalvelocity: V‖,e = Vz,0(1 − x3)3. The resulting curves are very much similar to those shown infigure 4, and therefore we do not report them here. As in the energy source, when a reversalin the safety factor is introduced, the source remains positive only in the outer region of theplasma, going to zero at the location where dqsaf/dr = 0. The absolute values in the negativeregion are seen to be larger compared with those obtained in the positive region. Similarlyto what we have done for the energy case, we have compared the momentum source withthe momentum flux V
ei . From equation (45) of [10], the latter reads (we rewrite also the
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
–40
–20
0
20
40
60
Mom
entu
m f
lux
and
sour
ce [
non-
dim
]
0.2 0.3 0.4 0.5 0.6 0.7 0.8x [non-dim]
Figure 6. The function inside the curly brackets of the momentum flux equation (39) and of themomentum source equation (40), for the regular safety factor profile (continuous and dashed lines,respectively) and for the strongly reversed safety factor profile (boxes and circles, respectively).The dotted line at zero is for reference.
momentum source for comparison) as
(rB0t)UVei = 3
a2
Lei
NeMeNeV‖e
− 5
L2q
− 1
Lq
[1
LNe
+1
LV‖e
+1
LNi
(1 + 2ηi
)], (39)
(1
L
)V
ei = 3
aL
Lei
NeMeNeV‖e
− 1
Lq
−[
1
LNe
+1
LV‖e
+1
LNi
(1 + ηi
)]. (40)
As usual, an over-hat indicates normalization. In figure 6 we plot the functions inside the curlybrackets of (39) and (40) for the case of regular and strongly reversed safety factor profile.Even though the curves reported here are very similar to the corresponding curves relativeto the energy case presented in figure 5, the actual values are somewhat different. As a finalpoint, we note from V‖e −j‖/(eNe) that the momentum source would produce an anomalouselectrical conductivity contribution, j‖/σ‖an, as well as a term ∝ dj‖/dr , in Ohm’s law. Astudy of the generalized current equation obtained with the results presented in this paper andin [10] will be presented in a forthcoming paper.
7. Summary and conclusions
This work has focused on the turbulent sources present in the parallel momentum and in theenergy transport equations, and originating from the resonant interaction between particlesand turbulent fluctuations in a high-temperature, axisymmetric, magnetized plasma. Theadopted theoretical framework is the action-angle transport formalism. First, the origin ofthe sources has been elucidated by presenting a detailed derivation, under the ansatz ofa drifting-Maxwellian closure, of the general transport law starting from the action-anglecollision operator which retains the friction term (contrary to what is done in the quasilinearapproach). Second, a detailed calculation of the sources has been presented, which clarifies theadopted mathematical simplifications and the corresponding physical implications, and thusthe limitations built in the final results. For example, we have incorporated in the formalism an
19
Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
ansatz for the magnetic turbulent spectrum, which retains the thermal structure of the spectrumfound in the original collision operator, but replaces its form with a model spectrum that betterrepresents the spectra of realistic experiments. While this step allows for an explicit evaluationof the sources, a more rigorous turbulent calculation would be required to obtain more reliablequantitative predictions, and to obtain truly self-consistent results.
The expressions for the momentum and energy sources derived in this paper pertain tocollisionless transport of passing electrons in steady-state magnetic turbulence. These sources,always present when the plasma has a supra-thermal level of fluctuations, depend on the densityand temperature equilibrium profiles of both electrons and ions, and on the first and secondpowers of the inverse scale length of the safety factor profile. The energy source includes anadditional term proportional to the factor Ti/Te −1, so that both its magnitude and sign dependon the relative values of the electron and ion temperature. In particular, this term (which is theonly surviving one in a truly Maxwellian plasma) leads to an energy flux that flows from thehotter species to the cooler species, a property that a quasilinear approach would have missed.
As an application of the theory, we have examined the influence of different shapes ofthe safety factor profile on the sources, since this novel dependence is probably the mostinteresting output of the analysis. Note that this dependence has been found in turbulent fluxesas well, both under the assumption of turbulent equipartition, and from the same action-angletransport formalism adopted in this work. A numerical plot of the energy and momentumsources for typical tokamak equilibrium profiles and for a level of magnetic perturbationconstant across the plasma minor radius (and assuming Ti = Te) shows the important impactof the safety factor profile on the magnitude and sign of the sources. In particular, the sourcescan have both positive and negative sign depending on whether the safety factor profile ismonotonically increasing toward the edge of the plasma (as in conventional discharges), orpresents a minimum inside the plasma (as in reversed magnetic-shear discharges). Utilizingresults previously published by the authors, we have verified that the contribution to the energyand momentum evolution coming from the flux and from the source can be of the same order ofmagnitude. Finally we note that, in a self-consistent numerical study, the turbulent generationof momentum would act back on the original equilibrium velocity profile, modifying both itsintensity and the location of its steepest gradient. This effect could be relevant to the transitionbetween different confinement regimes.
In conclusion, the expressions for the turbulent sources presented in this work, andthe preliminary numerical studies on their dependence on the plasma profiles, indicate theirpotentially important role in determining the evolution of the equilibrium profiles of turbulentplasmas, and thus their potential impact on the global stability properties of a plasma discharge.The dependence on the safety factor profile is a particularly interesting output of the analysis,and it would be interesting to know whether it could be recovered following more conventionalapproaches to transport. Finally, in order to assess in a more quantitative way the role playedby turbulent sources such as those derived in this work, numerical studies carried out withtransport codes incorporating the sources should be performed. By such a practice, one wouldalso remove our simplifying assumption of a constant level of magnetic turbulence.
Acknowledgments
One of the authors (IC) thanks the Italian Foreign Affair Ministry for the financial supportduring his doctoral studies, and the ENEA-Frascati Research Center for the hospitality offeredduring the last two years of his doctoral program.
This work was also supported by the Italian Ministry of Education, University andResearch.
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Plasma Phys. Control. Fusion 51 (2009) 075002 I Chavdarovski and R Gatto
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