Full orbit simulation of collisional transport of impurity ions in the MAST spherical tokamak

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Plasma Phys. Control. Fusion 53 (2011) 054017 (15pp) doi:10.1088/0741-3335/53/5/054017

Full orbit simulation of collisional transport ofimpurity ions in the MAST spherical tokamak

M Romanelli1, K G McClements1, J Cross2, P J Knight1, A Thyagaraja1

and J Callaghan3

1 EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB,UK2 Department of Physics, University of Bath, Bath BA2 7AY, UK3 Department of Physics, University of Oxford, New College, Oxford, UK

E-mail: michele.romanelli@ccfe.ac.uk

Received 7 September 2010, in final form 11 February 2011Published 7 April 2011Online at stacks.iop.org/PPCF/53/054017

AbstractTransport analysis of MAST discharges indicates that collisions are animportant loss mechanism in the core of a tight aspect ratio tokamak. In thestrongly varying equilibrium fields of MAST many of the assumptions of driftkinetic and neoclassical theory (e.g. small plasma inverse aspect ratio and lowratio of toroidal Larmor radius to poloidal Larmor radius) are not met by allparticle species and it becomes appropriate to use full orbit analysis to evaluateheat and particle fluxes. Collisional transport of impurity ions (C6+ and W20+)has been studied using a full orbit solver, CUEBIT, to integrate the test-particledynamics. Electromagnetic fields in MAST plasma have been modelled usingthe cylindrical and toroidal two-fluid codes CUTIE and CENTORI. A detailedstudy of the scaling of the test-particle diffusivity with collisionality in theequilibrium field reveals deviations from the standard neoclassical theory, inboth the Pfirsch–Schluter and banana regimes, and difficulties in defining alocal diffusivity at low collisionalities. The effect of electric and magneticfluctuations is also briefly addressed. It is found that field fluctuations enhancethe non-diffusive nature of transport. The full orbit analysis presented herepredicts levels of transport and confinement times for the examined speciesbroadly consistent with the experimental observations.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Transport analysis of plasmas in the MAST spherical tokamak [1] show that, while electronconductivities are largely ‘anomalous’ all over the discharge, ion conductivities in the plasma

0741-3335/11/054017+15$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1

Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

core (over up to 60% of the minor radius) have values not far from those predicted by thestandard neoclassical theory [2–4]. This experimental evidence suggests that collisionaltransport can be an important loss mechanism in spherical tokamaks and that it is necessary tore-examine the fundamental assumptions of the neoclassical theory, in light of the parameterspertaining to this configuration. Indeed collisional transport in spherical tokamaks, wherethe toroidal component of the confining magnetic field is of the same order of the poloidalcomponent at the low-field side, is not expected to be correctly described by the standardneoclassical ordering [5] based on an expansion in the drift parameters (ε = a/R � 1;Bθ/Bφ � 1 where a and R are the plasma minor and major radii, and Bθ and Bφ the poloidaland toroidal components of the confining magnetic field, respectively). Previous analyticaland numerical results have highlighted the fact that tight aspect ratio tokamaks can exhibitnon-local collisional transport [6] along with deviations from the neoclassical ordering thatcan only be captured by a full orbit particle analysis [7]. Moreover, in the presence of toroidallyasymmetric time-dependent field fluctuations (turbulence) neither toroidal angular momentumnor particle energy is conserved, and for fluctuations with sufficiently short wavelength themagnetic moment may cease to be an adiabatic invariant. In view of the considerations above,in order to compute accurately the transport of heat and particles in the fluctuating fields of aspherical tokamak it is appropriate to go beyond the drift kinetic theory and integrate the fullparticle orbits. In this framework we have studied the transport and confinement of impurityions in the MAST spherical tokamak with a full orbit particle code CUEBIT [8] coupled totwo global electromagnetic turbulence and transport codes CUTIE [9] and CENTORI [10].

The CCFE spherical tokamak MAST has major radius R = 0.85 m and minor radiusa = 0.65 m, hence inverse aspect ratio a/R ≈ 0.8. The typical toroidal magnetic field on themagnetic axis is 0.6 T and the typical plasma current is 1 MA. The main heating system is neutralbeam injection, with currently powers up to about 3.5 MW achieved. The beam introduces alarge population of fast deuterons into the plasma (with concentration nfD/ne ∼ 10% where nfD

indicates the density of the energetic deuterium andne the electron density) with nominal energypeak at 70 keV. Carbon and oxygen are the most abundant impurities with concentrations aroundnC/ne ∼ 5% and nO/ne ∼ 1%, respectively. The ion temperatures achieved routinely in theplasma core are of the order of 1 keV. With the above plasma parameters typical values of theLarmor radii of deuterons (thermal, energetic), carbon ions and oxygen ions are, respectively,ρD ∼ 1.0 cm, ρfD ∼ 8 cm (where 40 keV deuterium ions have been assumed), ρC,O ∼ 2.5–3.0 cm. At mid-radius (r/a = 0.5 where r is a radial position ranging between 0 and a) in atypical MAST discharge, the ion orbit widths for these species are of the same order as theLarmor radius. The typical gradient scale-lengths of temperature, density and safety factor areLT,n,q = 0.3 m at mid-radius, dropping to around 0.2 m at r/a = 0.8 [11].

The paper is organized as follows: in the next section the particle following code CUEBITis briefly described, along with the scheme used to couple the code with those providing theequilibrium and fluctuating electromagnetic fields. In section 3 results on carbon collisionaltransport in the large aspect ratio limit (benchmark with analytic theory) and in MASTstationary equilibrium magnetic field are presented, and in section 4 the effects of theequilibrium electric field and electromagnetic field fluctuations on impurity transport are brieflydiscussed. Our conclusions are presented in section 5.

2. Particle orbit integration

Impurity transport has been investigated numerically with CUEBIT by initializing a populationof test particles on a given magnetic surface and following their full orbits in the prescribedequilibrium and fluctuating electromagnetic fields, taking into account collisional interactions

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

with a flowing background ion species. We will assume that the measured transport of the testparticles representing an impurity species is representative of the transport of that species withinthe background plasma. This method can be regarded as a numerical simulation of a beaminjection or laser blow off experiment, which can be used to obtain direct information on particletransport [12]. The CUEBIT code is used to solve the Lorentz force equation for chargedparticles in arbitrary time-varying electric and magnetic fields using an algorithm that ensuresconservation of total energy to machine accuracy when E = 0 [13]. The equilibrium magneticfield expressed in terms of a poloidal-flux function ψ(R, Z) and a toroidal magnetic field alongwith the equilibrium electric field, plasma velocity and field fluctuations are calculated usingthe two-fluid transport and turbulence codes CUTIE (cylindrical, circular equilibrium) andCENTORI (fully toroidal, shaped magnetic surfaces). For the case in which E is a potentialfield, the total energy of collisionless particles is generally conserved to a satisfactory level ofaccuracy by taking the time step to be around a tenth of a Larmor period. In the laboratoryframe the Lorentz force equation then takes the form

mdv

dt= Ze(E + v × B) − m

τ(v − V ) + mar. (1)

The electric and magnetic fields as well as the bulk ion fluid velocity V include boththe equilibrium (slowly varying) and the fluctuating (rapidly varying in time and space)components:

E = E0(ψ, t0) + δE1(ψ, θ, ϕ, t1); B = B0(ψ, t0) + δB1(ψ, θ, ϕ, t1);V = V0(ψ, t0) + δV1(ψ, θ, ϕ, t1).

In the above expressions we have adopted toroidal coordinates with toroidal angle ϕ andpoloidal angle θ . The equilibrium fields are taken to vary on the energy confinement timescale while the rate of variation of the fluctuating components is assumed to be of the orderof the ion diamagnetic frequency (10–500 kHz). The use of different time variables t0 and t1in the above expression reflects these different timescales. The flux surface averages of therapidly varying components of the fields are generally small compared with the equilibriumfields; when this condition is broken, a transition to a new equilibrium may occur and thelarge fluctuation becomes part of the new equilibrium (examples include L–H transitions, ITBformation, sawtooth crashes). The parameter τ = τ(ψ, t1) is the slowing down collisiontime (in the case of trace impurities it is assumed to be velocity independent while dependingon the equilibrium plasma parameters) and ar(ψ, t1) = (arx, ary, arz) is a set of randomnumbers, chosen independently for each particle and at each time step, with zero mean andstandard deviation equal to the thermal velocity of the background plasma ions divided bythe geometric mean of the collision time and the time step δt used to follow the particleσ = (2Ti(ψ, t1)/m)1/2/

√τδt [8].

For trace impurities with thermal velocities much smaller than the background ion thermalvelocity the scattering process is isotropic in velocity space and the average scattering rateis well approximated by the slowing down frequency. The presence of noise terms in thethree components of equation (1) ensures that collisional pitch-angle scattering is taken intoaccount. For the case of a Maxwellian distribution of field particles the collisional slowingdown frequency (τ−1) of an impurity with the background ions is given by the expression

1

τ= m

1/2i

m

Z2Z2i e

4ni(ψ, t1) ln

6√

2π3/2ε20T

3/2i (ψ, t1)

, (2)

where the index i denotes the background species, e is the proton charge, ln is the Coulomblogarithm, T denotes the temperature and ε0 is the permittivity of free space. Note that in the

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

absence of fluctuating fields the test particles are interacting with the background plasma boththrough the collision frequency (which depends on position) and through the plasma velocityand equilibrium electric field. The latter is taken to be the electric field which confines thebackground ions, as calculated from the ion momentum-balance equation

E0(ψ, t0) = mi

eV0 · ∇V0 +

1

nie∇pi0 − (V0 × B0) + ηj0, (3)

where j0 is the equilibrium current density. The last term on the right-hand side of equation (3)is associated with the toroidal component of the equilibrium electric field Eϕ = ηjϕ , and isresponsible for the Ware pinch.

The two-fluid global electromagnetic turbulence codes CUTIE and CENTORI describethe nonlinear saturated phase of short wavelength drift-wave instabilities (driven by density,velocity and temperature gradients) in interaction with longer wavelength resistive-MHDmodes (typical relative amplitude of fluctuations is δn/n ≈ 10−4). The codes calculatethe nonlinear spectrum of fluctuations with a typical resolution of m = 0–64, n = 0–32(poloidal and toroidal mode numbers, respectively) and 100 radial grid points. The time stepadopted in the codes is of the order of the Alfven time. The codes differ in the geometry, beingCUTIE circular cylindrical and CENTORI fully toroidal. The coupling of CUEBIT withCUTIE/CENTORI has been achieved both dynamically (i.e. with test-particle orbits beingcomputed simultaneously with fluctuating fields) and in post-processing mode. Since the testparticles do not affect the dynamics of the background plasma, the post-processing mode isthe most effective in terms of time required for running the simulations and it allows thepossibility of repeating the simulations to probe transport in different regions of the plasmaand with different particle species. In the post-processing mode CUTIE and CENTORI arerun separately, and the full 3D magnetic field, electric field and velocity field are stored inan intermediate file at simulated time intervals tC = 10/ωci (where ωci is the plasma ioncyclotron frequency). The file is then reduced by a postprocessor calculating at each time stepand toroidal position an expansion in terms of Chebyshev polynomials, which ensures the bestapproximation of the function for a chosen polynomial degree. A general field component a

is then expanded as

a[ψ(R, Z), θ(R, Z), ϕ, t1] =n∑

s,l=0

asl(ϕ, t1)Ts(R)Tl(Z), (4)

where the Chebyshev polynomials of the first kind are defined by the recurrence relation

T0(x) = 1; T1(x) = x; Tn+1(x) = 2xTn(x) − Tn−1(x). (5)

The coefficients asl(ϕ, t1) are stored in a data file. The maximum degree of the fittingpolynomial, n, is chosen accordingly with the number of grid points at which the field isgiven (depending on the numerical grid). Once the data file is ready, particle trajectories canbe computed with CUEBIT. At the beginning of the particle loop cycle, after initializing theparticle position and velocities, CUEBIT imports from the data file the coefficients asl(ϕ, t1)

for each field component at two successive times (t = 0 and t = tC). The trajectoriesof the particles are then calculated in the time interval tC with time step δt typically onetenth of the particle Larmor time (this might differ for each particle), where δt � tC. Thefields at each particle position and time step are computed by linearly interpolating in timeand toroidal angle the coefficients asl(ϕ, t1) in the polynomial expansion. Once the particleshave been advanced for a time tC the coefficients for the next time interval are loaded andthe procedure is carried on till the end of the simulation. As discussed in the next section,the number of iterations necessary for calculating the local diffusion coefficients varies with

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collisionality. Particles reaching 95% of the plasma minor radius are counted as lost andremoved from the simulation. The particle confinement time can be calculated by followingthe test particles until their total number has dropped by a factor of e. Confinement times inMAST conditions are typically a few tens of milliseconds. The most demanding CUEBIT runsdiscussed here have been performed with a parallel version of the code, following the orbits of104 particles on 103 processors (10 particles per processor) for 50 ms and time step 10−8 s. Thepositions and velocities of the particles are periodically stored in an output file, together withstatistical information such as the average displacement of the ensemble of particles from theinitial magnetic surface and the mean square displacement. Diffusion coefficients, advectionvelocities and confinement times are then obtained by analysing the output, as explained inthe next section.

3. Collisional transport in the MAST tokamak

We have made use of two orthogonal coordinate systems: Cartesian coordinates (x, y, z);cylindrical coordinates (R =

√x2 + y2, ϕ = arctan y/x, Z = z where the axis of the

cylinder coincides with the torus symmetry axis); and the non-orthogonal toroidal coordinatesψ = ψ(R, Z); θ = θ(R, Z), ϕ = ϕ. The functions ψ, θ, ϕ will be referred to as flux function,poloidal angle and toroidal angle, respectively. In general the flux function ψ is chosen tobe negative at the magnetic axis and equal to zero at the plasma boundary. In the case of acircular cross section plasma a good choice of toroidal coordinates is ψ = ψ((R2 −R0)

2 +Z2),θ = arctan Z/(R − R0), ϕ = ϕ where the flux coordinate is a function of the minor radiusof the circular magnetic surface. Local transport analysis is carried out by positioning the testparticles on a chosen magnetic surface, denoted by ψ(R, Z) = ψ0, and at the same poloidaland toroidal angles θ0, ϕ0.

Thus the initial spatial distribution of test particles is given by

n(ψ, θ, ϕ, t0) = J0N0δ(ψ − ψ0)δ(θ − θ0)δ(ϕ − ϕ0), (6)

where J0 = |∇θ × ∇ψ · ∇ϕ|0 is the inverse of the element of volume in the chosen toroidalcoordinates calculated at ψ0, θ0, ϕ0. The three components of the initial velocities are chosenby sampling randomly a Gaussian distribution with zero mean and standard deviation equal tothe thermal velocity of the represented species (e.g. carbon). Alternatively, the particles canbe initialized with the same initial energy and a uniform distribution of randomly chosen pitchangles; after few collision times the particles are fully thermalized and have a Maxwelliandistribution with a temperature equal to that of the background ions. The particle trajectoriesare then calculated without collisions (i.e. with the two inertial terms on the right-hand side ofequation (1) switched off) for a few passing particle toroidal transit times or trapped particlebounce times, to allow the particles to become distributed all over their drift surfaces aroundthe initial magnetic surface (this process is better optimized when the particles are choseninitially to have the same energy). During this short initial time the particles drift from theinitial magnetic surface and position themselves on a range of drift orbits. This initial radialexpansion (here ‘radial’ refers to the direction locally perpendicular to the magnetic surface) isa ballistic process that must be filtered out from the diffusive and convective transport processesthat we aim at studying. In the absence of collisions and fluctuations the particles would remainon their drift surfaces and their average radial position would be constant in time. We definethe particle’s average position and mean displacement as follows:

r(t) = 1

N0

∑i

ri(t), r(t) =√√√√ 1

N0

∑i=1,N0

(ri(t) − r(t))2, (7)

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

R (cm)

Z (cm)

Figure 1. Close-up of the drift orbits of three particles on the low-field side of a MAST-like plasmaoriginating at the same point (black dot) on the magnetic surface located at R = 114 cm (black thickdashed line). Two of the particles (green and blue trajectories) are trapped and one (red trajectory)is passing.

Figure 2. Collisionless trajectories of 102 particles around the starting magnetic surface in aMAST-like equilibrium. The light (grey) curves indicate the plasma boundary.

where ri = ri(ψi(t)) = a√

1 − ψi(t)/ψaxis is the radial position (in length units) of the ithparticle and ψaxis is the value of the flux function at the position (R, Z) of the magnetic axis.Figure 1 shows the drift orbit of three particles in the initial distribution, computed withoutcollisions or electric fields, and figure 2 shows the distribution of particles around the startingmagnetic surface. Figure 1 illustrates the fact that, due to finite Larmor radius effects, thetotal orbital excursions of trapped ions in spherical tokamaks can be significantly greater thantheir guiding centre orbit widths; as noted by Gates and et al [7], this leads to an enhanceddiffusivity, for example, in the banana regime of neoclassical transport.

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

Successively the collision terms in equation (1) are turned on and the scattered trajectoriescalculated. In order to determine the scaling with collision time of the local ion diffusivityand convective velocity, the evolution of the test-particle distribution must be followed fora time tlocal that is short enough to ensure that the particles have not been transported toofar from the initial radial position, since the transport coefficient varies in space. On theother hand it is necessary to follow the trajectories for a time tυ that is long enough thatwell-defined transport coefficients can be inferred from spatial moments of the test-particledistribution; typically tυ must be at least several collision times [14]. The condition fortransport to be local (e.g. entirely ascribable to local parameters and field gradients) is therefore tυ � tlocal. When the collision time is chosen to be much shorter than the passing particlecirculation time or the trapped particle bounce time then the particles do not travel far fromtheir initial position on the magnetic surface and they are not subject to significant grad-B orcurvature drifts (responsible for the effects of toroidal geometry on collisional transport). Inthis case we recover classical transport. By increasing the collision time to a value longer thanthe average circulation time of passing particles but shorter than the average trapped particlebounce time, the test particles on average sample a significant fraction of the magnetic surfacebefore undergoing a collision, and the geometry (through the grad-B and curvature drifts)starts to influence the diffusivity. By increasing further the collision time to few bouncingtimes, a fraction of the test particles have closed banana orbits and the diffusion process isaffected by the appearance of radially extended orbits and collisional pitch-angle scatteringof particles close to the bounce points of trapped orbits (when the parallel particle velocitiesapproach zero). The scan in collision time described above has been performed by eliminatingfield fluctuations and the electric field and positioning 104 ions at r/a = 0.5 in both a CUTIEcircular equilibrium and in CENTORI elongated equilibrium. The use of a reduced numberof particles is allowed by the fact that the analysis is local in both radial position and time. Aconvergence study has been carried out by varying the number of particles between 103 and106. As expected the noise on the time dependence of the mean square displacement scalesinversely with the square root of the number of particles. The use of 104 particles is found toprovide a good estimate of the local transport coefficients for all the collision times analysed.In the rest of this section we will discuss four study cases: scaling of C6+ diffusion coefficientwith collision time in the large aspect ratio case small Larmor radius (benchmark of CUEBITwith analytical results); scaling of C6+ diffusion coefficient with collisionality in the largeaspect ratio case large Larmor radius (benchmark of CUEBIT with analytical results); scalingof C6+ diffusion coefficient with collision time in MAST equilibrium, small Larmor radius;scaling of C6+ diffusion coefficient with collision time in MAST equilibrium, large Larmorradius.

As a first step we have benchmarked CUEBIT against the neoclassical analytical resultsin the large aspect ratio limit. We have considered a low beta, circular cross section plasmawith inverse aspect ratio a/R0 = 0.1 (R0 = 10 m, a = 1 m). The toroidal component of themagnetic field is taken as Bϕ = B0R0/R, (B0 = 1.0 T) and the poloidal component of theequilibrium field is calculated for a plasma with q(a) = 4.0 (Ip = 121 kA, Bθ(0.5a) = 0.035 T,q(0.5a) = 1.4). The test particles (N0 = 104, A = 12, Z = 6, i.e. fully ionized carbon) areplaced initially at ri/a = r0/a = 0.5. Random toroidal angles (uniformly distributed) andconstant poloidal angle (equal to zero) are chosen as initial conditions; the initial particledistribution is thus n(r, t0) = (1/2πr0(R0 + r0))N0δ(r − r0)δ(θ). This choice of initialpositions and velocity distribution ensures that the fraction of trapped particles in the simulationis correctly represented (the fraction of trapped particles in the large aspect ratio limit isproportional to the square root of the inverse aspect ratio). The gyrofrequency and Larmorradius of each individual particle depend on its initial position and perpendicular velocity. For

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

Figure 3. Computed diffusivity (diamonds) versus collisionality in the classical (a) and P–Sregimes (b) (ρ = 0.7 × 10−3 m, Vth = 3.4 × 104 ms−1, R0/a = 10) and analytical Dcl and DPS(circles).

the chosen magnetic field the gyrofrequency ranges from ωci = 4.5 × 107 s−1 on the low-fieldside to ωci = 4.9 × 107 s−1 on the high-field side (this variation can be neglected in the largeaspect ratio limit). The average Larmor radius is ρ = 0.7 × 10−3 m. Particle trajectories,ri(t), θi(t), ϕi(t), are calculated solving the stochastic equations (1) described in the previoussection for a time exceeding several collision times τ (typically 50 τ ).

Particles are transported radially by collisions. For short times such that r(t)/r0 � 1and (r − r0)/r0 � 1 the evolution of the radial distribution is described by

n(t, r) = N0

σ(t)√

2πe− [r−r(t)]2

2σ2 , (8)

where σ = √2D(t − t0), D being the diffusion coefficient appearing in the transport equation

∂n

∂t= D

∂2n

∂r2+ V

∂n

∂r, (9)

where V is a convective velocity. We can determine the diffusion coefficient from the particledistribution by computing r2 ≈ σ 2 = 2D(t − t0), with r defined in equation (7), whileany time variation in r indicates convection. Our first objective is to determine how D scaleswith the collision time τ in the given equilibrium magnetic field. As mentioned above, in orderto resolve the Larmor orbit we use a time step ten times smaller than the shortest cyclotronperiod. Different values of collisionality ν∗ = Rq/τVth are accessed by varying the collisiontime at constant thermal velocities Vth (and Larmor radii). We will adopt the conventionalterms of large aspect ratio neoclassical theory to define the different collisionality regimes:the banana regime ν∗ < ε3/2, the plateau regime ε3/2 < ν∗ < 1, the Pfirsch–Schluter (P–S)regime ν∗ ≈ 1 and classical ν∗ � 1. In order to access the banana regime with a thermalvelocity of the order of 104 ms−1 the collision time has to exceed 10−2 s. Having to computemore than 10 collision times requires the calculation to be carried out for a time between 0.1 sand 1 s while the P–S regime requires short simulations of less than 0.1 ms.

For the first case study (small Larmor radius, large aspect ratio R = 10 m, a = 1 m,A/Z = 2) we have chosen a thermal velocity of 3.4 × 104 ms−1 (corresponding to aLarmor radius ρ = 0.7 × 10−3 m) and varied the collision time by 7 orders of magnitude,with the smallest value equal to the cyclotron period. In figure 3 the diffusion coefficientscalculated with CUEBIT are compared at high collisionality with the expected classical values

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

Figure 4. Computed diffusivity (diamonds) versus collisionality in the P–S/classical (a) and banana(b) regimes (ρ = 0.01 m, Vth = 4.8 × 105 ms−1, R0/a = 10) and analytical DPS and DB (circles).

Dcl = 〈〈ρ2〉v〉θϕ/2τ , (where the two brackets indicate velocity space average and magneticsurface average 〈〈ρ2〉v〉θϕ = ρ2 = 2(mVth/ZeB0)

2 and Vth = (Ti/m)1/2) and at lowercollisionalities (ν∗ < 5) with the P–S diffusion coefficient DPS ≈ (1 + 2q2)ρ2/2τ , whereq = 1.4.

The divergence of the simulated and predicted diffusivities in figure 3(b) around ν∗ = 5indicates a transition from P–S to classical scaling. Banana regime scaling is expected forν∗ < ε3/2 = 0.01. However, for the chosen value of the Larmor radius the expected diffusioncoefficient in the banana regime (DB ≈ ε−3/2q2ρ2/2τ ) is of the order of 10−4 m2 s−1 orsmaller and would require long simulation with a larger number of particles (lower noise) toresolve it properly. We will therefore discuss the banana regime in the next case study wherea Larmor radius ten times larger has been chosen. For all values of collisionality used inthe first case study the relative displacement of the average of the particle distribution wasnegligible, between 10−3 and 10−2. For the second case study (larger Larmor radius, largeaspect ratio, A/Z = 2) we have repeated the scan above for ions having a thermal velocityof 4.8 × 105 ms−1, corresponding to an average Larmor radius of 0.01 m (0.0106 m on thelow-field side, 0.0097 m on the high-field side). The collision time required to access thebanana regime for this choice of parameters is one order of magnitude smaller (4 × 10−3 s)and the collisionality regime can be accessed promptly with a smaller number of iterations.The scaling of the calculated diffusion coefficient with collisionality in the P–S/classical andbanana regime is reported in figure 4.

Figure 4(b) shows the scaling of the diffusion coefficient with collisionality in the bananaregime (ν∗ < 0.01). The order of magnitude of the diffusion coefficients agrees with theanalytical prediction. It is important to note that the analytical scaling represents an asymptoticbehaviour and does not describe the transition regions between different collisionalities (whereterms ordered small in the drift kinetic equation become of the same order of leading terms).The roll over observed at ν∗ = 0.005 is not described by the asymptotic behaviour captured bythe neoclassical theory and arises from being close to the transition between the banana regimeand the plateau regime. Indeed a non-monotonic behaviour is found in this region. Moreover,in the low collisionality regime we observe also a collisional radial-drift (pinch velocity) VcD

of the peak of the particle distribution which scales with the square of the Larmor radius. Thecomponent parallel to the magnetic surface of the curvature and centrifugal drifts producesa radial drift through the drag term of equation (1) (VcD = −m(v∇B × B/ZeB2)/τ where

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

(a)

q = rBT (Rout) / BpRout

qRoutRin R

ε = 0.8

ε = 0.8

ε = 0.1

Z

JG10

.393

-3c

Figure 5. (a) Cross sections of three flux surfaces having the same value of qout but different aspectratios or elongation. (b) Magnetic field lines on the magnetic surfaces in (a) for ε = 0.1 dottedline) and ε = 0.8 plotted in geometric angles. The field structure is chosen to coincide aroundθ = 0 and θ = 2π .

v∇B = ρVth∇B × B/B2; defining 1/LB = |∇B × B/B2| we thus have |VcD| = ρ2/LBτ )that quickly takes the particles away from the initial surface, well before diffusion begins tomodify the particle distribution. This effect is shown in figure 8. Therefore, the transport at lowcollisionality is intrinsically non-diffusive and non-local. The diffusion coefficients reportedin figure 4(b) have been computed by linearly interpolating r2(t) between 30 and 50 collisiontimes. For the third case study we have repeated the collisionality scan for a tokamak withlarge inverse aspect ratio, a = 0.65 m, R0 = 0.85 m, B0 = 0.5 T and small ion Larmor radius(ρ = 1.4×10−3 m the thermal velocity being 3.4×104 ms−1). The calculated diffusivities havebeen compared with the large aspect ratio limit neoclassical diffusion coefficients. Specifically,the neoclassical diffusion coefficients are those of particles having the same Larmor radius andcollision time but placed in the field of a conventional tokamak on a surface having the samevalue of the cylindrical q as that calculated at the low-field side of the tight aspect ratio plasma(qout = rBT(Rout)/BpRout) under study, as illustrated in figure 5 for the extreme cases ε = 0.1and ε = 0.8

In figure 6 the mean square displacement and the evolution of the radial distributionof the test particles initially located at mid-radius (r0 = 0.325 m) are shown for three highcollisionality cases.

The scaling of the diffusion coefficient with collisionality for this third case is shown infigure 7.

Lowering the collisionality from ν∗ ≈ 10 to ν∗ ≈ 1 (the P–S regime) and plotting themean radial displacement versus time, we find that two linear phases appear: initially theparticles diffuse classically but later the diffusivity increases. Figure 7(a) shows the scalingwith collisionality of the diffusion coefficient in the second diffusive region (diamonds) ascompared with the reference P–S coefficient defined above (dots). The computed diffusivityfound in the PS regime is lower than that calculated taking the actual value of q at the initialmagnetic surface (q = 1.4).

At the lowest collisionalities, figures 7(b) and (c), the effect of the finite orbits onthe diffusive transport becomes more important and deviations from neoclassical diffusion

10

Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

800

600

400

200

0

1000

20

(a)

40 60 800 100

<∆r

2 > (m

m2 )

τ = 1x10-8sτ = 1x10-7sτ = 1x10-6s

Time (µs)JG

10.3

93-1

a

1600

1200

800

400

0

2000

0.315

(b)

0.320 0.325 0.330 0.3350.310 0.340

Num

ber

of p

artic

les

1τ10τ20τ30τ40τ50τ

Radial position (m)

JG10

.393

-1b

Figure 6. Mean square displacement of 104particles (Vth = 3.4×104 ms−1) for three high collisionfrequencies (a) and evolution of the radial particle distribution with τ = 8 × 10−6 s (b).

apparent. Figure 8 shows the mean square radial displacement versus time for collisiontimes 10−3 s, 2 × 10−3 s and 3 × 10−3 s. The mean square displacement does not increaselinearly, hence the transport is non-diffusive; however, we can define an effective diffusioncoefficient by linearly interpolating the slowly varying part of the curves. At this level oflow collisionality, unlike in the high collisionality case, the effective diffusion coefficient canhave a non-monotonic dependence on collisionality as seen in figure 7(c). Also shown infigure 8 is the evolution of the particle distribution for a simulation with τ = 10−3 s. A clearinward pinch was observed throughout the simulation; after 50 collision times the peak ofthe distribution had moved 55 mm inwards from the initial position. It was observed that theGaussian distribution became skewed towards the magnetic axis. The particles were spreadradially over 0.25 m after a time of 50 ms, thus they travelled much further in 50 collision timesthan at higher collisionality.

In the fourth case study we have calculated the transport of particles under the sameconditions as case three, but with a thermal velocity of 4.8 × 105 ms−1, corresponding toρ = 0.02 m, and initial positions r0/a = 0.5 and r0/a = 0.8. The results confirm thoseillustrated in figure 8, namely non-diffusive transport at low collisionality dominated by astrong inward convection of particles. When initialized at r0/a = 0.8 many particles are lostfrom the plasma within 50 collision times.

4. Effects of radial electric field and field fluctuations on particle transport

In the presence of field fluctuations (e.g. turbulence) it is no longer possible to define localtransport coefficients, and particle transport becomes highly non-diffusive and non-local,subject as it is to particle advections in fields that are non-uniform in both space and time[15, 16]. Figure 9 shows the scattered trajectory of a test particle in MAST fields, simulatedwith CENTORI (m = 0–64, n = 0–32 corresponding to k⊥ρi ∈ [ρi/a, 1] and typicallyδn/n ≈ 10−4 at saturation), followed until it is lost at r/a = 0.95.

As a test case we have considered here the effect of fluctuations on the confinement ofW20+ impurity ions in MAST conditions. The W ions were placed on the magnetic axis ofan H-mode MAST-like rapidly rotating plasma with parameters close to those of shot number

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

Figure 7. Scaling of the calculated diffusion coefficients (diamonds) with collisionality in theP–S (a) and banana (b), (c) regimes (ρ = 0.0014 m, Vth = 3.4 × 104 ms−1, R0/a = 1.25).Circles indicate the values of the analytical neoclassical P–S diffusion coefficient (a) and DB (b),(c) calculated with q = qout and ε = 0.4. The squares in (a) indicate the values of the P–S diffusioncoefficient calculated taking q = 1.4.

#18220 (Te,i = 0.6 keV, ne = 5 × 1019 m−3, τ = 2.5 × 10−5 s, ρ = 5.3 × 10−3 m) andfollowed in the equilibrium and fluctuating fields. Particles which reached 95% of the minorradius were regarded as lost and removed from the simulation. If the transport were purelydiffusive, the mean radial position of the set of particles would evolve on timescales much lessthan the confinement time according to the expression

r = 2π

N0

∫ ∞

0r2n(t, r) dr =

√πDt.

As a first step we have analysed, in the above conditions, the collisional transport of W ionsin the absence of fluctuations and radial electric fields. We followed the time evolution of themean position for approximately 25 ms. The W ions were transported on average a distanceof 0.25 m from the magnetic axis (40% of the minor radius), and r exhibited a clear t1/2

dependence, full line in figure 10(a), indicating a diffusivity of 1.1 m2 s−1 [15]. This is closeto the expected P–S value in the plasma core, and implies a neoclassical confinement time ofaround 400 ms. As shown in figure 10(b), full line, only a small fraction of the particles werelost in this simulation.

Adding the equilibrium radial electric field and fluctuations for MAST shot #18220 andrepeating the simulation we find that the W20+ ions are transported much faster as a result

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

0.001

0

0.002

0.003

10

(a)

20 30 400 0

<∆r

2 > (m

m2 )

τ = 1x10-3sτ = 2x10-3sτ = 3x10-3s

Time (ms)

JG10

.393

-2c

1600

1200

800

400

0

2000

0.175 0.225 0.275 0.325 0.375

(b)

Num

ber

of p

artic

les

1τ10τ20τ30τ40τ50τ

Radial position (m)

JG10

.393

-2b

Figure 8. Mean displacement of 104 particles from the mean radial position in time for three lowcollision frequency cases (a) and evolution of the radial particle distribution with collision time10−3 s (b).

R (cm)

Z (cm)

Figure 9. Particle orbit in MAST #18220, with fields calculated using CENTORI.

of the combined effect of collisions, E × B drifts and fluctuating fields (at the time of thisinvestigation fluctuating fields computed using CENTORI were not available and we have thusused CUTIE data for the case with fluctuations). As shown in figure 10(a) the W ions are onaverage transported across 95% of the plasma minor radius in 25 ms under the effect of thestationary E ×B drift only (figure 10(a), dotted line) and in only 10 ms when fluctuations areincluded (not shown). It is apparent from figure 10 that the transport process is non-diffusivewhen the stationary and fluctuating E × B drift is taken into account.

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Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

no E-field

with E-field

no E-field

with E-field

Figure 10. Mean position of 104 W20+ ions in #18220 MAST H-mode plasma after 25 ms frominjection (a). The equilibrium and radial electric field were calculated using CENTORI. Thecontinuous line is calculated without including the radial electric field and the dotted line with theradial electric field. (b) Fraction of confined W20+ ions versus time when only collisional effectsare accounted (continuous line) and including the equilibrium electric field (dotted line).

The confinement time of W20+ in MAST #18220 is estimated by counting the lost particlesand is found to be 16 ms. This is in agreement with the estimate of the energy confinement timein the CUTIE simulation (17 ms) and with experimental measurements of carbon confinementtimes.

5. Conclusions

The results of a numerical study on full orbit calculation of collisional impurity transport inMAST equilibrium fields have been presented. The effect of field fluctuations on impuritytransport has been briefly discussed in section 4. The stochastic trajectories of a set of testparticles have been calculated in the equilibrium and simulated fluctuating electromagneticfields of the MAST spherical tokamak using a full orbit code (CUEBIT). The fields havebeen computed with two independent two-fluid turbulence codes CUTIE and CENTORI. Thetest particles have been initialized on different magnetic surfaces and their local diffusioncoefficient and convective velocity calculated from the time evolution of the radial distributionfunction. First CUEBIT was benchmarked for the case of collisional transport in the magneticfield of a large aspect ratio tokamak and ion Larmor radius normalized to the plasma minorradius ρ/a < 10−3. Numerical results for the scaling of diffusivity with collisionality havebeen shown to be generally in quantitative agreement with the predicted neoclassical scaling.Finite Larmor radius effects become important at low collisionalities for large ρ/a. It hasbeen found that at the lowest collisionalities and for larger Larmor radii the advective velocityarising from the magnetic field non-uniformity coupled with collisions moves the particlesaway from the initial magnetic surface before diffusion begins to dominate the process (non-diffusive transport). In this context it is therefore in general not possible to define a strictly localdiffusion coefficient for the more energetic ions even in a large aspect ratio tokamak. In theabove case the fluxes have to be determined by a spatial average of the driving gradients in theneighbourhood of the initial magnetic surface. Finite orbit effects are found to be important forthermal and energetic particles in the confining magnetic field of a tight aspect ratio tokamakat low collisionalities, where diffusion coefficients larger than neoclassical are found. At thelowest collisionalities advection dominates over diffusion and transport is non-diffusive andnon-local. Strong advective velocities arise when the full fluctuating electromagnetic field of

14

Plasma Phys. Control. Fusion 53 (2011) 054017 M Romanelli et al

MAST is introduced in the problem. The effect of the electric field and fluctuations on the globalconfinement of impurities has been studied in the case of tungsten ions in a typical H-modeMAST discharge (shot #18820) where impurity transport is found to be strongly enhanced bythe presence of MAST large radial electric field (consistent with the strong plasma rotationvelocity), and the confinement time is reduced by a factor of about 20 by the combined effect ofthe equilibrium radial electric field and the fluctuating fields. In conclusion full orbit calculationof ion transport accounts for corrections over the neoclassical level of transport for thermal andenergetic ions in a tight aspect ratio tokamak and energetic ions in a conventional tokamak.When the full equilibrium electric field, plasma velocity and electromagnetic fluctuations aretaken into account the full orbit calculation produces particle confinement times in the rangeof those measured in the experiment.

Acknowledgments

This work was funded by the United Kingdom Engineering and Physical Sciences ResearchCouncil under grant EP/G003955 and the European Communities under the contract ofAssociation between EURATOM and CCFE. The views and opinions expressed herein donot necessarily reflect those of the European Commission.

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