Full orbit simulations of collisional impurity transport in spherical tokamak plasmas with strongly-sheared electric fields
aa r X i v : . [ phy s i c s . p l a s m - ph ] J un Full orbit simulations of collisional impuritytransport in spherical tokamak plasmas withstrongly-sheared electric fields
C G Wrench , E Verwichte and K G McClements Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, CV47AL, UK EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon,Oxfordshire, OX14 3DB, UKE-mail: [email protected]
Abstract.
The collisional dynamics of test impurity ions in spherical tokamakplasmas with strongly-sheared radial electric fields is investigated by means of a testparticle full orbit simulation code. The strength of the shear is such that the standarddrift ordering can no longer be assumed and a full orbit approach is required. Theeffect of radial electric field shear on neoclassical particle transport is quantified fora range of test particle mass and charge numbers and electric field parameters. Itis shown that the effect of a sheared electric field is to enhance the confinement ofimpurity species above the level observed in the absence of such a field. The effectmay be explained in terms of a collisional drag force drift, which is proportional toparticle charge number but independent of particle mass. This drift acts inwardsfor negative radial electric fields and outwards for positive fields, implying stronglyenhanced confinement of highly ionized impurity ions in the presence of a negativeradial electric field.PACS numbers: 52.55.Fa, 52.25.Vy, 52.25.Fi, 52.65.-y
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Nuclear Fusion ollisional impurity transport and sheared electric fields
1. Introduction
Impurity ions present in tokamak plasmas can significantly degrade plasma performancethrough radiative losses and fuel dilution [1]. Conversely, impurity ions present in theedge or near the divertor of a tokamak may be beneficial by radiating thermal energy andthereby mitigating the heat load incident on the divertor and plasma facing components[2]. An understanding of impurity transport (a topic which has received less attentionthan the transport of bulk ions) is crucial in order to predict the overall performance offusion plasmas [1].The analysis of charged particle motion in strongly-magnetized plasmas, forexample those in tokamaks, generally relies heavily on a multi-timescale approach [3],whereby the particle motion is decomposed into fast gyromotion about the magneticfield, streaming motion along the field (characterised by adiabatic invariance of theparticle magnetic moment) and a much slower drift motion of the particle guiding centreacross the field. This treatment is valid provided that the drift ordering is satisfied.Specifically, the particle Larmor radius must be small compared to the length scales onwhich the electric, E , and magnetic, B , fields vary, the cyclotron period must be shortcompared to the field variation time scales and the E × B drift velocity must be smallcompared to the particle thermal speed.In spherical tokamaks (STs), such as the Mega Amp`ere Spherical Tokamak (MAST)[4], the drift ordering is often not applicable, and it is then not possible to treat particlemotion analytically. On the low field side of a ST plasma the poloidal component ofthe magnetic field can be comparable to the toroidal component, so that the drift orbitwidth of a trapped particle can be of the same order as the particle Larmor radius.Moreover, in ST plasmas with edge transport barriers (ETBs) the associated radialelectric field can vary on length scales comparable to both the drift orbit width and theLarmor radius of thermal ions [4].It has been shown that both collisionless trapped particle guiding centre orbits andparticle gyro-motion are significantly distorted by strongly-sheared radial electric fields[5, 6]. This effect is often referred to as orbit squeezing [7], although in fact the widthsof trapped particle orbits can be either reduced or increased, depending on the signof the electric field gradient. The effects of orbit squeezing on neoclassical bulk iontransport have previously been studied in the limit of large aspect ratio [5, 8] using thisframework. However, the analytical theory of orbit squeezing assumes linearly shearedelectric fields, and recent measurements of radial electric fields associated with ETBs inspherical tokamaks indicate field profiles that are not consistent with the assumption ofconstant shear [4]. This highlights one limitation of the current theory and motivatesthe use of a numerical approach.We note that non-uniformly sheared electric fields have previously been investigatedusing a full orbit approach [9]. However, this study investigated the impact of shearon particle E × B flow and only considered collisionless particle orbits. In the presentwork we investigate the impact on the motion and collisional transport of impurities of ollisional impurity transport and sheared electric fields
3a strongly-sheared electric field in a MAST-like spherical tokamak equilibrium using afull orbit, test-particle simulation code. The fields are assumed to be axisymmetric andstatic; we therefore neglect turbulent transport effects.
2. Model
The transport of impurities in a spherical tokamak with a prescribed local radial electricfield is simulated using the test particle code CUEBIT [10, 11], which solves the Lorentz-Langevin equation m z d v d t = Ze ( E + v × B ) − m z τ zi ( v − u ) + m z a r , (1)where m z , Ze and v are the impurity mass, charge and velocity, respectively. Theterm on the right-hand side of (1) that is proportional to v − u models the frictionfrom the bulk ions. In the absence of an electric field it ensures that the impuritiesrelax to a drifting Maxwellian distribution temperature T i and flow u matching thoseof the bulk ions. Here, u is taken to be zero; we assume that the radial Lorentzforce is sub-dominant to the pressure gradient term in the radial component of thebulk ion fluid momentum balance equation. This assumption is consistent withreported measurements of temperatures, densities and flows in the vicinity of ETBsin MAST [4, 12]: the pressure gradient is typically several tens of kPam − whereas thecontributions of poloidal and toroidal flows to the radial component of the Lorentz forceare typically around one or two kPam − at most. In taking u to be zero we also neglectany deviation of the bulk ion distribution from a stationary Maxwellian arising fromneoclassical effects [13]. The quantity τ zi is the classical impurity-bulk ion collision timegiven by [13] τ zi ≡ ν zi = 6 √ π / ǫ ln Λ m z T / m / Z e n i , (2)which is a function of the local bulk ion temperature and density profiles, T i and n i ,respectively. Here, ln Λ is the Coulomb logarithm and ǫ is the permittivity of free space.The final term on the right-hand side of (1) models random accelerations ofimpurities due to Coulomb collisions with bulk ions. The vector a r is a set of threerandom numbers, chosen independently for each particle at each time step from aGaussian distribution of zero mean and variance σ = 2 T i ν zi m z ∆ t , (3)where ∆ t is the time step that is used in the numerical simulation. Although theinstantaneous collisions are taken to be isotropic, the cumulative effect of these overan impurity Larmor orbit naturally reflects the effects of gradients in the bulk iontemperature and density profiles (leading to, for example, thermal forces), which leadsto anisotropy in the collisions. Here we have again assumed that the bulk ions have aMaxwellian distribution. ollisional impurity transport and sheared electric fields Z n z ≪ n i ( m e /m i ) / ,where Z is the impurity ion charge number, n z is the impurity ion number density, n i is the bulk ion number density and m e and m i are the electron and bulk ion massesrespectively [13]. This condition essentially ensures that collisions of bulk ions withimpurity ions are sufficiently infrequent that they have a negligible effect on the bulkplasma neoclassical transport and automatically ensures that collisions between impurityions can be neglected.A MAST-like equilibrium is considered, which is modelled using an analyticalSolov’ev-type solution of the Grad-Shafranov equation with a linear pressure profileand a potential toroidal magnetic field component. In cylindrical coordinates (R, φ ,Z),the poloidal flux, Ψ, is of the formΨ = Ψ (cid:26) γ (cid:20)(cid:16) R − R (cid:17) − R (cid:21) + 1 − γ R Z (cid:27) , (4)with R = 0 .
964 m, R b = 0 .
93 m, γ = 0 . ≃ . − corresponding toa total plasma current of 1 MA. The magnetic field vector is calculated from (4) as B = RB φ ∇ φ + ∇ Ψ × ∇ φ where RB φ = 0.386 Tm. We assume T i and n i are flux functionsand prescribe analytical forms, T i = T ΨΨ m + T , (5) n i = n ΨΨ m + n , (6)where Ψ m the flux at the magnetic axis and T = 1 keV, T = 0 . n = 5 × m − and n = 1 × m − . These profiles are broadly consistent with those measuredin co-rotating MAST H-mode plasma discharges [14]. In such discharges, the effectiveion charge is often measured to be close to unity throughout the core plasma, indicatingno significant impurity accumulation.A local radially sheared electric field E = −∇ Φ near the edge is included, with anassociated electrostatic potential Φ of the formΦ (Ψ) = Φ arctan (cid:18) Ψ − Ψ ∆Ψ (cid:19) , (7)where Φ , Ψ and ∆Ψ are the potential barrier height, location in Ψ space and widthin Ψ space, respectively. Measurements reported by Meyer et al [4] of edge radialelectric fields in MAST H-mode plasmas (see for example, figure 7 of [4]) show thatthe corresponding potential profile is approximately of the form given by (7) and thatthese constants have typical values of Φ =60 V and ∆Ψ=1.35 10 − Tm . We choose,for computational ease, to centre the electric field structure on Ψ =0.6Ψ( R , . For our particular choice of equilibrium, these quantities correspond to aphysical location and width in the outboard Z =0 plane of R =1.2 m and ∆ R =1.1cm. The corresponding peak electric field strength, E , is − . − , which is closeto measured values in MAST ETBs [4]. We shall present results from simulationswith varying values of Φ and ∆Ψ within the ranges − . ≤ Φ ≤ −
120 and0 . R , ≤ ∆Ψ ≤ . R , ollisional impurity transport and sheared electric fields E φ = − . − . Includingthe 1 /R dependence of this field has a small, O( <
3. Characterising transport
In the absence of an electric field we expect neoclassical particle transport to be purelydiffusive [13]. For tokamak plasmas with slowly-varying profiles, transport coefficientsmay be deduced empirically from moments of the spatial distribution of test particles[16, 17]. Figure 2, in which the evolution of the mean minor radius of the impuritydistribution is plotted, illustrates a typical CUEBIT simulation in which 10 C ions areinitially released from the magnetic axis in the absence of any electric field. Overlaid is a t / fit expected from diffusive transport [18]. This corresponds to a diffusion coefficientof D ≈
10 m s − , which is close to the expected Pfirsch-Schl¨uter neoclassical diffusivityin the plasma core, D ∼ q ρ /τ zi . Here q is the safety factor and ρ L is the Larmorradius.In the case of narrow transport barriers, with plasma properties varying on lengthscales down to the Larmor radius, it is not possible to infer local transport coefficientsusing this method. In such cases a local effective diffusivity D eff may be deduced directlyfrom the local radial particle flux Γ z and density gradient ∂n z /∂ρ : D eff = − Γ z ∂n z /∂ρ . (8)Here n z is the flux surface-averaged minority ion density and ρ = ρ (Ψ) is a radialcoordinate, defined as the flux surface averaged minor radial coordinate ρ (Ψ) = Z ΨΨ m R Rdl/ |∇ Ψ ′ | R Rdl d Ψ ′ , (9)with l the arc length along a flux surface in the R–Z plane. ρ (Ψ) is evaluated numericallyas a function of Ψ for the Solov’ev equilibrium described above. To quantify the impactof sheared electric fields on particle transport the value of D eff is normalised to its valuemeasured in the absence of any electric field, D .Clearly D eff is a simplified formulation of the particle diffusion coefficient since weassume that particle flux scales linearly with the minority ion density gradient. However,we wish to emphasise that Γ Z , which appears in (8), is the full particle flux as computedby CUEBIT. This includes the effects of transport driven by bulk ion gradients, e.g. thethermal force arising from the bulk ion temperature gradient (see, for example, equation5.9 of [13]). Such a force arises from the variation of the collision frequency across aLarmor orbit and, unlike to guiding centre and fluid calculations, appears naturally infull orbit particle simulations. ollisional impurity transport and sheared electric fields B = B z and E = E x bothassumed to be constant, where we use the x -direction as a proxy for the radial directionin tokamak geometry. Equation (1) can then be solved exactly, yielding v x = v ⊥ exp ( − ν zi t ) sin Ω t + E x B ν zi Ω 11 + ν / Ω , (10) v y = v ⊥ exp ( − ν zi t ) cos Ω t − E x B
11 + ν / Ω . (11)In tokamaks we have that ν ≪ Ω . In this limit, we are left with the usual E × B driftin the y -direction and an additional “drag force” drift in the x -direction, which arisesfrom the inclusion of a collisional drag in the test particle equation of motion. Thisimplies an inward-pinch in the presence of a negative radial electric field. Substitutingfor the impurity-ion collision frequency, given by (2), we find that the drag force driftcan be written as v d ≃ Z E x B n i T / m / e ln Λ6 √ π / ǫ . (12)This is independent of test particle mass but proportional to test particle charge.In order to relate the impact of a collisional drag force drift on particle transportwe consider the steady-state one-dimensional transport equationdd x ( D d n z d x + vn z x/ ∆ x ) ) = 0 , (13)where v is the peak value of the drag drift, v d . Here we have assumed a form of thespatial variation of E x that is consistent with the electrostatic potential given in (7),with x the displacement from the peak electric field ( x ≡ ¯Ψ − ¯Ψ ) and ∆ x a measureof the electric barrier width (∆ x ≡ ∆ ¯Ψ). We neglect all spatial variations in v d exceptfor that occurring due to its dependence on E x . We also assume that the impurity ionLarmor radii are small compared to ∆ x and that particle drifts in the x -direction aredue solely to the drag effect. The above form of the transport equation ensures thatthe test particle flux, which we assume is composed of the usual diffusive and advectiveterms, Γ z = − D d n z d x − v d n z , (14)is constant. Introducing a dimensionless spatial variable ξ = x/ ∆ x and a P´eclet numberPe = v ∆ x/D , (13) becomesd n z d ξ + Pe n z ξ = − γ, (15) ollisional impurity transport and sheared electric fields γ = Γ z ∆ x/D . This has the exact solution n z ( ξ ) = n exp (cid:16) − Pe tan − ξ (cid:17) " − γ Z ξ exp (cid:16) Pe tan − η (cid:17) d η (16)where n is the particle density at x = 0, i.e. at the position of peak electric field anddrag drift. For ξ ≪
1, or x ≪ ∆ x , this reduces to n z ( x ) = n exp (cid:18) − Pe x ∆ x (cid:19) − Γ z v (cid:20) − exp (cid:18) − Pe x ∆ x (cid:19)(cid:21) . (17)It follows that, at x = 0, we have D eff ≡ − Γ z dn z /dx = D vn / Γ z . (18)Identifying v with the drag force drift velocity, and assuming that n z / Γ z is essentiallyconstant between simulations, we therefore find that the local effective particle diffusioncoefficient scales with test particle charge as D eff = D αZ , (19)where α = E x B n n i T / m / e ln Λ6 √ π / ǫ Γ z , (20)and is independent of both the test particle mass and charge. Despite theapproximations used to derive (19), we will show in the next section that it provides afairly accurate description of impurity ion transport at an ETB in full toroidal geometry.
4. Impurity transport simulations
We now present simulation results for a number of impurity species. In table 1 we list,for reference, the normalised collision frequency, ν ∗ ≡ qRτ zi v th , (21)of each species at the position of the simulated transport barrier. Here v th is the particlethermal velocity. The quantity ν ∗ measures the number of collisions a particle undergoesin one toroidal transit of the tokamak, with ν ∗ ≤ ν ∗ > ions of various impurityspecies were released from the magnetic axis in equilibria with and without an inward-directed, radially sheared electric field. The electric field acts to confine C ions. Thecomparison of density profiles of C [see dashed lines, figure 3(a) and figure 3(b)] showsa step in the density profile at a minor radius of approximately 0.4 m. This indicatesthat the sheared electric field is acting as a barrier to the outward radial transport ofthe ions. The location of this step coincides with the peak of the radial electric field. ollisional impurity transport and sheared electric fields Table 1.
Normalised collision frequency for impurity ions simulated with computedeffective diffusion coefficients, (8), without, D , and with, D eff , a sheared radial electricfield as given by (7). For those simulations indicated by ∗ we have modified the locationof the barrier, such that Ψ = 0 . R ,
0) = − . , and temperature and densityprofiles, such that T = 0 .
75 keV, T = 0 .
05 keV, n = 5 × m − and n = 2 × m − .Impurity species ν ∗ D D eff He . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ∗ Ar . ± . . ± . . ± . . ± . ∗ Mo . ± . . ± . . ± . . ± . ∗ W . ± . . ± . . ± . . ± . ∗ W . ± . < . The variation of impurity density with minor radial coordinate for three otherimpurity species, He , Ne and W , are shown on figure 3. The density profilesare similar in the absence of an electric field but differ substantially when it is present.The greatest confinement occurs for W . All impurity species simulated have reachedthermal equilibrium with the bulk ions particles before they begin interacting with theradial electric field. The poloidal distribution after 100 ms of simulated time are shownin figure 4 for He , Ne and W ions, which again clearly illustrates the enhancedconfinement of impurity ions by a sheared radial electric field, particularly in the caseof W .In order to compare the scaling of effective diffusivity with particle parameters,given by (19), with simulations we consider results for a number of test particle specieswith varying mass and charge numbers. In each case the simulation is run until theratio given by (8) at the peak of the electric field has reached a steady-state, except fornoise fluctuations arising from the use of a finite number of test particles. However, theabsence of a continuous particle source means that both the particle flux and the densitygradient continue to slowly decline. In figure 5a the normalised diffusion coefficient isplotted for particle mass numbers in the range 20 (neon) to 184 (tungsten) all withZ=10. We see that the diffusion coefficient is essentially independent of mass number.In figure 5b the test particle mass number is held fixed at 20 but the charge state isvaried from Z=2 to Z=10. Equation (19) predicts that the quantity D/D eff − ollisional impurity transport and sheared electric fields D/D eff − Z . Furthermore, inderiving (19) contributions to the particle flux from bulk ion density and temperaturegradients were neglected (cf. (14) and equation 5.9 of [13]). Including such terms wouldadd to (19) a term that is non-linear in Z in its denominator and which could explainthe possible turn-over present for high Z in figure 5. Figure 6 shows the variation with electric field width and height of the effective diffusioncoefficient for C ions. The normalised effective diffusivity has a power law scaling withelectric barrier width for constant barrier height. The coefficients of the fit themselvesscale linearly with barrier height. Thus the scaling of diffusion coefficient with electricfield width and strength may be expressed in the form D eff = D (cid:20) ∆ R ∆ R (cid:21) E/E m s − , (22)where ∆ R is the FWHM of the electric field in the outer midplane and E is the peakheight of the radial electric field barrier. We see that the diffusion coefficient scalesas the width of the electric field barrier to a power of the electric field strength, withthe width of the barrier normalised to a fraction of the bulk ion Larmor radius. Thisscaling has been verified for ∆ R and E in the ranges 0 . ρ Li ≤ ∆ R ≤ ρ Li and − . − ≤ E ≤ − .
65 kVm − for all impurity species, with ρ Li the bulk ion Larmorradius. In the limit of no electric field, this scaling recovers the effective diffusivity asfound in simulations without an electric field barrier. The scaling parameters for variousspecies are listed in table 2.Unlike (19), the empirical expression for D eff given by (22) does in fact depend onthe barrier width, implying limits in the validity of the approximations used to derivethe former. We expect that for large ∆ R/ρ Li , the effective diffusion coefficient will nolonger have a power law scaling but will tend asymptotically to a result similar to (19).We note, from table 2, that ZE and ∆ R are broadly constant. Therefore, thedependency of impurity transport on the shape of the electric field barrier dependsapproximately only on the charge state of the species, not on the species mass. Thus,we can approximate this scaling as D eff ≃ D ∆ R . ρ Li ! ZE/ . (23)
5. Discussion
We have simulated realistic radial electric field profiles for ETBs in a spherical tokamakequilibrium and investigated the impact on collisional impurity transport of the strengthand width of the electric field profile for a variety of impurity species. We havedemonstrated that full orbit simulations provide a valuable insight into the transportof particles in such scenarios. It has been shown that a strongly-sheared radial electric ollisional impurity transport and sheared electric fields Table 2.
Scaling of computed effective diffusion coefficient, (8), with electric fieldstrength, E , and width, ∆ R , for several impurity species.Impurity species D (m s − ) ∆ R E (kVm − )C . ± . . ± . ρ Li . ± . . ± . . ± . ρ Li . ± . . ± . . ± . ρ Li . ± . . ± . . ± . ρ Li . ± . field can significantly increase the confinement of test particle ions. This effect has beenexplained in terms of a drift arising from the drag between impurity and bulk ions, withthe direction of the drift given by the radial electric field and its strength determinedby the impurity charge. Clearly, the negative implication of this study is the enhancedconfinement of highly ionized impurity species in the vicinity of strong, negative radialelectric fields.In our derivation of the drag force drift given by (12) we neglected both spatialvariations in the radial electric field and toroidal geometry. However, the scaling ofeffective diffusivity with Z inferred from using this drift velocity in a simple slab modelof the transport barrier, (19), is in agreement with our test-particle simulations in whichno approximations are made with regard to finite Larmor radius or toroidal effects.This suggests that test-particle transport in the immediate vicinity of the barrier is notstrongly-dependent on the effects of toroidal geometry or the collisionality regime (i.e.the value of ν ∗ ). This further indicates that finite Larmor radius effects are relativelyunimportant. One would expect electric field shear on length scales approaching theparticle Larmor radius to result in a modified drag force drift velocity, analogous to the E × B drift with finite Larmor radius correction discussed by Tao et al [9]. However,this requires further study.The location of the electric field in the simulations, which resembles more an internaltransport barrier than an ETB, was chosen in order to lower computation time byreducing the distance from the plasma core to the location of the electric field, therebyreducing the time before particles begin interacting with the electric field. This alsohas the benefit of improving particle statistics, since the particles are less dispersedradially and the number of particles at the barrier is consequently higher. Simulationsin which the electric barrier is located closer to the plasma edge, listed in table 1,demonstrate that moving the barrier closer to the edge has little qualitative impact onthe results obtained. Furthermore, the variation of collisionality of impurity species,with normalised collisionality ranging from 0 .
014 (banana regime) to 4 .
74 (Pfirsch-Schl¨uter regime), appears to be unimportant, since the measured normalised diffusivityremains independent of mass and scales as expected from the drag force drift argumentwith particle charge.Whilst the effect of bulk plasma rotation was neglected in the present study, onewould expect transport barriers to be associated with flows of the bulk ions (e.g. zonal ollisional impurity transport and sheared electric fields to undergo very rapid collisional transport (due to thecombined effect of trapping in a centrifugal potential well and a modification to theeffective magnetic field arising from the Coriolis force). One would therefore expecta competition between this enhanced transport due to rotation and the improvedconfinement resulting from sheared radial electric fields.We note that the effect of improved confinement due to a sheared radial electricfield is more important for spherical tokamak plasmas than for conventional tokamaks,primarily due to the strong B dependence in (12). For example, in the MAST-likeequilibrium used throughout, we have that the Ware pinch velocity is of the order v Ware ≃ qE φ /ǫB ∼ . − and the drag force drift velocity is of the order v d ∼ . − for C (where we have used B = 0 . E x = −
10 kVm − , E φ = − . − , n i = 3 × m − , T i = 500 eV, q = 3 and ǫ = 0 . v Ware ∼ . − and v d ∼ .
02 ms − (assuming B = 3 T, E x = 20 kVm − , E φ = − .
04 Vm − , n i = 3 × m − , T i = 5 keV, q = 3 and ǫ = 0 . Acknowledgments
This work was funded by the RCUK Energy Programme under grant EP/G003955 and aScience and Innovation award, and by the European Communities under the contract ofAssociation between EURATOM and CCFE. The views and opinions expressed hereindo not necessarily reflect those of the European Commission.
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Nuclear Fusion Plasma PhysicsControlled Fusion Phys. Plasmas R ad i a l e l e c t r i c f i e l d [ k V m − ] ( s o li d li ne ) S quee z e f a c t o r ( da s hed li ne ) Figure 1.
Form of the prescribed electric field (solid line) and corresponding squeezefactor for a C (dashed line) ion in the outer midplane of a MAST-like equilibrium.The full-width half-maximum of the radial electric field is chosen to be equal to thebulk ion gyroradius evaluated at the location of maximum electric field. The squeezefactor is calculated from S = 1 + Ze Φ ′′ ( r ) /m z Ω . ollisional impurity transport and sheared electric fields −3 M ean r ad i a l d i s p l a c e m en t [ m ] Figure 2.
Mean minor radial coordinate versus time of C ions released at themagnetic axis in a simulation with zero electric field (squares). Overlaid is a fitted t / curve expected of diffusive transport (solid line). P a r t i c l e den s i t y [ m − ] Minor radius [m] (a) P a r t i c l e den s i t y [ m − ] Minor radius [m] (b)
Figure 3.
Density profiles of impurity ions against minor radius, both without (a) andwith (b) a sheared radial electric field. Simulated ions are He (——), C (– – –),Ne (— · —) and W ( · · · · · · ). Vertical lines in (b) indicate the peak strength andfull-width half-maximum positions of the sheared radial electric field. ollisional impurity transport and sheared electric fields V e r t i c a l po s i t i on [ m ] (a) (c)(b) Figure 4. (Colour online) Poloidal distribution of (a) He (b) Ne and (c) W ions in a MAST-like equilibrium with a radially sheared electric field, the position ofwhich is indicated by a dashed (red online) line. ollisional impurity transport and sheared electric fields D e ff / D (a) Mo Ar Ne W D / D e ff − (b) Figure 5.
Scaling of particle diffusion coefficient with particles parameters. (a) scalingwith test particle mass with constant charge number Z=10. (b) scaling with testparticle charge for constant mass number of 20 (neon). ollisional impurity transport and sheared electric fields Peak electric field strength [kVm −1 ] E l e c t r i c f i e l d w i d t h [ m ] −7−6−5−4−3−2−1000.010.020.030.040.050.060.070.08 00.10.20.30.40.50.60.70.80.9 Figure 6.
Contours of constant normalised effective diffusion coefficient for C6+