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Kinetic Monte Carlo simulation of escaping core plasma particles to the scrape-off layer for
accurate response of plasma-facing components
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2013 Nucl. Fusion 53 073023
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IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION
Nucl. Fusion 53 (2013) 073023 (8pp) doi:10.1088/0029-5515/53/7/073023
Kinetic Monte Carlo simulation ofescaping core plasma particles to thescrape-off layer for accurate responseof plasma-facing componentsV. Sizyuk and A. Hassanein
Center for Materials under Extreme Environment, School of Nuclear Engineering,Purdue University, West Lafayette 47907, USA
E-mail: [email protected] (V. Sizyuk) and [email protected] (A. Hassanein)
Received 15 October 2012, accepted for publication 13 May 2013Published 11 June 2013Online at stacks.iop.org/NF/53/073023
AbstractDuring normal and disruptive operations in tokamak devices the escaped core plasma particles are a potential threatto the divertor and nearby component lifetime as well as plasma contamination. Comprehensive enhanced physicaland numerical models are developed and implemented in the upgraded High Energy Interaction with GeneralHeterogeneous Target Systems (HEIGHTS) package to accurately predict the impact of the escaped particles onplasma-facing and nearby components. An ab initio Monte Carlo-based kinetic model of the escaping core particlesis developed for integration with the magnetohydrodynamic (MHD) models of the initiated edge plasma where theescaping particles are used as an input volume source. The paper describes details of the 3D Monte Carlo kineticmodel, validation and benchmarking and simulation results for both National Spherical Torus Experiment and ITERdevices using actual reactor design and magnetic configurations. The simulation results are being implementedself-consistently with various HEIGHTS models that include surface erosion, divertor plasma generation, plasmaMHD evolution, heat conduction and detailed photon transport of line and continuum radiation.
1. Introduction
Successful development of future fusion reactors criticallydepends on the correct prediction of the heat and particleloads to reactor walls and the optimum material choice forplasma-facing components (PFCs). Experimental data ofcurrent tokamak devices and theoretical predictions show thatfuture ITER-like surfaces should be resistant both to steady-state heat flux of up to 20 MW m−2 and to intense transient(τ = 0.1–10 ms) events of up to 10 GW m−2. The divertorplate is a key component of the tokamak device. Highparticle and heat fluxes can cause high surface sputtering,melt and vaporization erosion, plasma contamination andcomponent failure that can interrupt normal operation of thereactor. The reactor geometry, the magnetic configurationand the interaction processes in the scrape-off layer (SOL)determine not only heat transport and loads to componentsurfaces but also play an important role in plasma confinementand control the transition to the high-confinement mode (H-mode) operation. Experiments in current tokamak devicesshow the complex behaviour of edge plasma interaction withdevice components and as a result the required understanding
of the complex self-consistent behaviour of plasma in theentire SOL configuration. The edge plasma interaction withcomponent surfaces affects the behaviour of deuterium–tritium(DT) bulk plasma and impurity contaminations, influencestoroidal plasma motion and redistributes energy loads amongreactor components. Damage of plasma-facing and structuralmaterials due to loss of plasma confinement remains one ofthe most serious concerns for safe, successful and reliablereactor operation. Plasma instabilities occur in variousforms such as hard disruptions, which include both thermaland current quench, edge-localized modes (ELMs), runawayelectrons and vertical displacement events (VDEs). Therefore,accurate understanding of the escaping core plasma into SOLand the subsequent interaction with device components isextremely important for successful development, operationand optimization of future fusion devices. Current fusionreactors have complex 3D design and simple one- or two-dimensional models cannot be used for correct description ofself-consistent processes of edge plasma evolution. Transportof charged particles in tokamak plasma environment is bestdescribed by kinetic theory [1]. However, solving the fullkinetic equations remains an impractical task and most recent
0029-5515/13/073023+08$33.00 1 © 2013 IAEA, Vienna Printed in the UK & the USA
Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
investigations [2–5] are based on the magnetohydrodynamic(MHD) approach for modelling the core plasma expansion intothe SOL area. Simple estimation, for example of the NationalSpherical Torus Experiment (NSTX) tokamak pedestal plasmaconditions [6], i.e. for ne = 7 × 1013 cm−3, Te = 500 eVand Ti = 750 eV, showed that the time between scatteringevents [7] for ions is τi = 4 × 10−4 s, and for electrons isτe = 4 × 10−6 s. The predicted parameters for ITER pedestalplasma (ne = 3 × 1013 cm−3, Te = 1.7 keV and Ti = 2.5 keV[8]) give even more pessimistic numbers that prevent thecorrect use of the Navier–Stokes equations: τi = 1×10−2 s andτe = 2×10−4 s. The classical MHD approach assumes that thetime between particles scattering is much smaller than durationof the simulated events (duration of the ELM time is aboutτELM = 10−4–10−3 s [9]). However, as was shown in recentstudies [2–4], implementation of various modifications andassumptions to the basic MHD system of equations allows oneto calculate the parameters of the escaped core plasma particlesclose enough to the measured experimental data. The mainproblem lies in the fact that transport properties of the tokamakmagnetized plasma cannot accurately be described with pureclassical theory. The classical transport theory is valid forcollisional plasma in a homogeneous and stationary magneticfield [10]. The classical theory should be expanded to includevarious turbulence mechanisms to explain the anomalousplasma transport properties in tokamaks [11].
The simulation results were significantly improved whenthe general SOL transport problem was divided into twosubtasks: (1) MHD of the escaping core plasma, starting fromthe last closed flux surface (LCFS) and (2) MHD of the SOLneutrals (background gas) [3]. Solution of the first subtaskfor the escaping core plasma is used as a volume source forthe neutral transport equations. This approach was used ina simplified NSTX geometry where the profiles of divertorheat and particle fluxes were calculated and compared withexperimental data [12]. Recent investigations showed thatsplitting of the SOL transport problem and implementationof various hybrid approaches are prospective ways for bettermodelling of edge plasma evolution and correctly predictingcomponent heat loads [13–16]. However, it should be statedthat a full direct integrated model of the SOL plasma isstill a very complex problem and only simplifications allowone to simulate characteristics of tokamak operation in areasonable time.
Most recent experimental and theoretical studies ofPFC erosion only considered plasma behaviour around localareas of the tokamak geometry [17–20]. Investigators havesignificant limitations in computational power and relevantexperimental data due to design complexity, the numerousphysical processes involved and difficulties in the integrationof micro- and macro-scale processes. However, understandingvarious edge plasma processes and their interdependence in theentire SOL area are critically needed for accurate and reliableoperation of future higher power devices. In our previousstudies [18], we simulated the evolution of edge plasmaduring normal and disruptive operation only in localizedareas of the SOL using the High Energy Interaction withGeneral Heterogeneous Target Systems (HEIGHTS) computersimulation package with various integrated models [21–23].These simulations showed a potential serious damage of nearby
divertor surfaces in ITER-like devices during giant ELMsand disruption, not directly from the core plasma particlesbut from the high radiation fluxes of the evolving divertorplasma-shielding layer. We simulated in detail the energy loadof core plasma particles deposited into component surface,heat conduction, melting and vapourization of the divertormaterial. The vapour MHD evolution and expansion wasmodelled including vapour heating by the incoming escapedcore plasma particles and photon radiation generated within.Magnetic diffusion, heat conduction and radiation transportwere also studied in the generated edge plasma. We concludedthat the erosion of divertor surfaces is a complex self-consistentproblem where the heat load to localized areas should becoupled through MHD and radiation transport processes ofthe evolving shielding cloud. Expansion of the computationaldomain to include the entire SOL area leads, however, to thenecessary multiscale description of the complex reactor designthat results in unavoidable extensive calculation time and thecomplex description of calculation domain. For example, thesize of a typical fusion device is measured in metres, accurateMHD calculations require cell size of 100–200 µm, and detailsurface erosion processes require a fine discretization level,with cell size less than 0.5–1.0 µm.
In this study, we introduce for the first time, acomprehensive kinetic Monte Carlo model for the simulationof the escaping core plasma and for the prediction of heat loadto tokamak PFCs using the entire 3D device geometry. Weimplemented detail description of the escaping core particlesinto the SOL that will be used as volume source input in theHEIGHTS integrated models. This work includes (1) a modelof the core particles escaping and a benchmark with availabledata and (2) simulation results that show details of the escapedparticle drift separation and composition on inner and outerdivertor surfaces. Based on the above-mentioned advantagesof splitting the transport problem, our upgraded model includesboth of the two separated subtasks. However, opposed toother existing models, we do not use any fitting parameters oraverages from continuum media, i.e. we do not solve Navier–Stokes equations for the escaped core DT plasma. Similar tothe UEDGE model [12], we separate the escaping core plasmaparticles and the edge plasma evolution; however, opposite tothe UEDGE model we simulate the motion of the escaping coreplasma as individual particles in the complex electromagneticfield structure of the device. The calculations presented belowshow the comparison between both models. However, theadvantages of our approach are the ability to study various driftphenomena, to separately divide the contribution of ions andelectrons into component heat loads and to exclude artificialfitting parameters usually used in the description of anomaloustransport properties. The particle’s motion is calculated usingab initio models with simulation of all possible scatteringprocesses along their path. Results from our developed MonteCarlo kinetic model will then be used as an input volume sourcefor the solution of the second subtask, i.e. MHD evolutionof the edge plasma. It should be stated that the completesolution should include the edge plasma MHD ‘feedback’, i.e.SOL hydrodynamics will influence particles escaping, driftand scattering processes as well. Initially SOL is filled withnear vacuum gas density. The scattering probability of theD, T and e in SOL is therefore small. As consequences
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Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
of plasma instabilities and interactions with divertor/wall,surface vaporization causes a density increase in SOL. As aresult, the escaped core particles will undergo scatterings inSOL far from device walls. We are currently extending ourmodels to two-component mixture plasma (DT + impurities)in the next HEIGHTS package upgrade. Future studies willtreat SOL with real ‘dirty’ edge plasma, calculation of thethermodynamic properties of partially ionized plasma andsimulation of radiation transport processes in mixed multi-component plasmas. The main goal of this study is to validateour approach of splitting the SOL transport problem and toassess the applicability of the developed kinetic Monte Carlomodel for the calculation of the volume input source for theedge plasma MHD equations. We analysed and compared oursimulation results with known experimental data.
2. Mathematical and physical models
As we discussed in our previous work [18], the edge plasmaMHD equation system is described using the conservation lawsof mass, momentum, energy and magnetic field as
∂ρ
∂ t+ ∇ · (ρv) = Qm,vap
∂ρv
∂t+ ∇ ·
(ρvv + pt − BB
4πµ
)= − 1
4πµB(∇ · B)
∂et
∂t+ ∇ ·
[v(et + pt) − 1
4πµ(v · B) · B
+c2η
16π2µ2(∇ × B) × B
]
= Qth + Qrad + Qe,vap + Qimp − 1
4πµ(v · B)(∇ · B)
∂B
∂ t+ ∇ · (vB − Bv) +
c2
4πµ∇ × (η∇ × B) = −v(∇ · B).
(1)
Here, ρ is the density; v is the velocity; B is the magneticfield; et is the total energy, which includes the hydrodynamicpart, eh = ein + ekin and the magnetic part em = (B2/8πµ);ein is the internal energy; ekin = (ρv2/2) is the kineticenergy. Analogous to energy, pressure has hydrodynamic andmagnetic parts: pt = ph + (B2/8πµ). Magnetic diffusionprocesses are taken into account as the Joule heating term,(c2η/16π2µ2)(∇ × B) × B, in the total energy equation andas the diffusion term, (c2/4πµ)∇×(η∇×B), in the magneticfield equation, where η is the resistivity, µ is the magneticpermeability and c is the light speed. In our calculations weassume µ = 1 for the plasma. The thermal conduction in theplasma is included in the energy equation as the Qth term. Thetarget vapourization processes are taken into account as Qm,vap
and Qe,vap sources in the continuity and energy equations,respectively. The Qrad term describes the radiation transportprocesses. The external energy source Qimp that takes intoaccount the energy input from the escaped core particles intothe edge plasma is the main theme of this study.
To avoid the limitations described above in edgeplasma modelling and to preserve the multiscale approachin the integrated physical and mathematical models, we
Figure 1. Schematic illustration of the quadtree meshe refinementmodel to describe NSTX geometry.
implemented the adaptive mesh refinement (AMR) algorithmwith unstructured grid geometry [24] and upgraded allintegrated routines in the HEIGHTS numerical package andcomputational algorithms for the AMR configuration. A five-layer quadtree method [25] is used for the AMR simulationof the entire SOL plasma evolution (figure 1). The AMRtechnique was integrated with our algorithms for the MPIparallel implementation of HEIGHTS.
The equilibrium plasma configuration used in our study forthe initial conditions was extracted from the EQDSK databasefiles [12, 26] and reconstructed for the quadtree mesh (figure 2).The magnetic field components were calculated from the storedpoloidal fluxes as
Br = 1
r
∂�
∂z, Bϕ = RcBc
r, Bz = −1
r
∂�
∂r. (2)
Here, Br,ϕ,z is the magnetic field components in any pointP(r, ϕ, z); � is the poloidal magnetic field and Rc, Bc arethe radius and toroidal magnetic field at the tokamak magneticcentre.
The RBBBS and ZBBBS values stored in the standardG-EQDSK file1 determine the tokamak core plasma border,i.e. the LCFS where we begin sampling the start point ofthe escaped core particles. HEIGHTS has the flexibility todescribe the escaping particles from the LCFS using any spatialand temporal distribution to cover the possible wide range ofescaping core plasma parameters during normal and disruptiveoperations. We can also use different initial temperaturesof electron and ion compositions. In this study, we usedequal probability distribution of the escaped particles along theentire LCFS and uniform temporal distribution of the impact
1 https://fusion.gat.com/theory/Efitgeqdsk
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Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
Figure 2. Initial magnetic field, separatrix and LCFS locationsreconstructed from the EQDSK file (ITER design).
Figure 3. Schematic illustration of the escaped core particles in the3D Monte Carlo package.
power deposition. The initial direction of the particles wassampled as 3D equiprobable so that electrons and ions can bereturned to the core plasma or can move along the ‘passed’ or‘trapped’ trajectory [27]. Figure 3 schematically illustrates oursimulation of the escaped core particle motion implemented inHEIGHTS.
As illustrated in figure 3, particles are moving in a 3Dcylindrical S(r, ϕ, z) coordinate system where r is the radial,ϕ is the toroidal and z is the axial coordinate. This is thegeneral system used in the HEIGHTS package. We do notuse the MC method for the solution of the continuity equation.The escaped core media are not considered as continuous withpre-calculated or measured transport coefficients along andacross the magnetic field. In our simulation, we considered realions and electrons with Larmour gyration and toroidal motionwithin the complex structure of the magnetic field. However,equations of motion of charged particles in the arbitrarydirected magnetic and electric fields are extremely awkward.We, therefore, introduced and expressed the equation of motionin the auxiliary Cartesian system S ′(x ′, y ′, z′), where the z′ axisis the direction parallel to the local magnetic field vector �B, i.e.in our virtual coordinate system S ′, there is only one magnetic
field component B ′z = | �B| = B. This redefinition allows
expressing the equation of motion in a compact form for thevelocity components:
V ′x =
(V ′
0x − cE′y
B
)cos
(qB
mrct
)
+
(V ′
0y − cE′x
B
)sin
(qB
mrct
)+
cE′y
B
V ′y =
(V ′
0x − cE′y
B
)sin
(qB
mrct
)
+
(V ′
0y − cE′x
B
)cos
(qB
mrct
)− cE′
x
B
V ′z = V ′
0z +qE′
z
mt,
(3)
and for coordinates:
x ′ = x ′0 +
cE′y
Bt +
mrc
qB
{(V ′
0x − cE′y
B
)sin
(qB
mrct
)
−(
V ′0y +
cE′x
B
)cos
(qB
mrct
)+ V ′
0y +cE′
x
B
}
y ′ = y ′0 − cE′
x
Bt +
mrc
qB
{(V ′
0x − cE′y
B
)cos
(qB
mrct
)
+
(V ′
0y +cE′
x
B
)sin
(qB
mrct
)− V ′
0x +cE′
y
B
}
z′ = V ′0zt +
qE′z
2mr
t2,
(4)
where E′x,y,z are the electric field components in the S ′
coordinate system; x ′0 and V ′
0 are the initial location andvelocity of particle; mr = m/
√1 − V 2/c2 is the particle
relativistic mass; t is the time; and c is the speed of light.Equations (3) and (4) are valid within one MHD cell borderand one time step.
The transformation between the S and S ′ system can beexpressed using the three Euler angles: ϕ0, � and [28], whereϕ0 is the initial toroidal angle coordinate of the particle in theS system and the last two angles are determined from the localmagnetic field as
sin� = Brcosϕ0 + Bϕsinϕ0
|Bϕ| ,
cos� = Brsinϕ0 − Bϕcosϕ0
|Bϕ|
(5)
and
sin = |Bϕ|B
, cos = Bz
B. (6)
After calculation of the transformation matrices A and B:
A =[ cos� sin� 0−sin�cos cos�cos sin
sin�sin −cos�sin cos
],
B =[cosϕ0 −sinϕ0 0
sinϕ0 cosϕ0 00 0 1
], (7)
an image of vector V (r, ϕ, z) in a virtual system can becalculated as V
′ = ABV . Since both the initial and virtual
4
Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
systems are orthogonal, the reverse transformation can becalculated by simple matrix transpose V = BTATV
′.
The transport of individual particles in plasma isdominated by the long-range Coulomb force that results ina large number of small interactions. Since it is not feasibleto simulate each interaction, a single-collision Monte Carloapproach to particle transport is not applicable for manysituations of practical interest. Therefore, small interactionsare lumped together and treated in a continuous manner. In theinteractions of particles with walls, the energy losses are dueto soft interactions with the target atomic electrons (excitationand ionization loss). The changes in the electron direction aremostly due to multiple Coulomb scattering from the nucleus,with some contribution from soft electron scattering [29]. Dueto the limited computational power, it is difficult to take intoaccount all interactions of the incident particles with targetatoms. We divided these interactions into two groups: closecollisions and distant collisions as is traditionally used in otherMonte Carlo codes. The criterion for grouping is the energythat is transmitted to the recoiled particle. Since most of thedistant interactions do not have a critical effect on the incidentparticle energy or direction of motion, they can be consideredas the combined contribution of small interactions and theircommon effect can be comparable to the interaction of closescattering events, depending on the incident initial energy. Inour model, all interactions where incident particles lose <1%of their initial energy are considered as distant. The distantinteraction contribution (energy and direction change) isaccumulated along the trajectory between two close collisionsand is added at the end of the trajectory. This approach isapplied to the simulation of all scattering processes in theSOL and to tokamak component surfaces. Various physicalprocesses include ion–nuclear, ion–electron, electron–nuclear,electron–electron interactions, the bremsstrahlung process,photoabsorption, the Compton process and Auger relaxation.The algorithm takes into account contribution of the secondary(recoiled) particles, if their energy is higher than a criticalvalue. HEIGHTS models of plasma particle interaction withmatter are described in detail in [21, 29, 30]. The new 3DMonte Carlo algorithm is developed and benchmarked forplasma particle interactions with solid and plasma matter inthe magnetic field for any geometrical configuration.
3. Validation and benchmarking
The procedure of benchmarking HEIGHTS models andcomputer package includes both individual validations ofseparate blocks and components as well as benchmarking thewhole integrated package against available experimental andtheoretical data. In this study we present benchmarking resultsand comparisons that verify our new kinetic model for theevolution of the escaping core particles. To further demonstratethe applicability of our approach we performed benchmarksimulations for two different magnetic configurations, i.e.NSTX and ITER design. We simulated particle flow evolutionduring both normal and disruptive operations of plasma. Heatflux distribution on the divertor plate surface was the parameterused for comparison of results.
The NSTX (R = 0.85 m, a < 0.67 m, R/a > 1.27)
experimental and simulation data [12] was used to validate
Figure 4. Comparison of HEIGHTS calculated heat flux at theNSTX divertor surface with experimental data and other simulations.
our model in low-field devices. We used the published NSTXdischarge parameters for the initial conditions in our developedkinetic model. The temperatures used for the escaped coreelectrons and ions were Te = 130 eV, Ti ∼ Te − 15 eV.The incident power was estimated to be Pimp = 1.75 MW.The starting points of outgoing/escaping particles from thecore were assumed equally distributed around the LCFS andinitial particle velocity was directed in the 3D space with equalprobability. Figure 4 shows the simulated heat flux distributionon the divertor surface from our kinetic model in comparisonwith the heat flux measured in the NSTX experiment as wellas calculated by the UEDGE code [12]. Since HEIGHTScurrently calculates unperturbed escaped core particles, theflux distribution has a slightly narrower footprint in comparisonwith the experimental results. When taking into account edgeplasma evolution with impurity content, which is not correctlytreated in most previous simulations, it will result in broaderparticle/heat flux distributions and to the reduction of peaks.However, correct peak location, compliance of curves formand total energy demonstrate good agreement between ourcalculations and the published experimental data.
The second benchmark for model validation was per-formed for ITER geometry and magnetic field configurations.To confirm our previous calculations performed using local-ized geometry simulation [18], we compared the disruptionspatial power density footprints on the outer divertor sur-face. The initial conditions for disruption parameters in oursimulations assumed a total energy of the escaped particlesQ = 126 MJ, impact duration τ = 1 ms and particles tem-perature T = 3.5 keV. In previous calculations, the computa-tional domain was limited for the divertor nearby area; and theestimated spatial distribution [23] of disruption particles wassampled directly on the domain border. In our new model,the final spatial distribution is the result of the integrated ap-proach of the entire 3D SOL reactor geometry. A reasonableagreement between the two different models is illustrated infigure 5.
4. Simulation results
A full 3D kinetic description of the escaping core particles isrequired to realistically study particle evolution in the complexmagnetic configurations and geometries of current and futuretokomaks. We should stress that the presence of strong
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Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
Figure 5. Comparison of HEIGHTS 3D kinetic model withprevious localized model [18] calculations of the heat flux duringdisruption on the ITER outer divertor surface.
Figure 6. Heat flux footprint at the NSTX divertor surface. Powershot PNBI = 4.9 MW.
spatial effects cannot be described with simple and lowerdimensional models. The complex feature of the magneticfield-induced plasma drift in SOL and divertor nearby areasis the starting point to determine the hydrodynamics evolutionand radiation phenomena at the surface of PFCs. First, westudied the initial dependences that determine the heat loadconditions at the divertor surface. Based on our simulations,we should point out that the typical ratio of the outer to innerdivertor peak heat flux is based on the stochastic character ofthe escaping core particles in current device magnetic fieldconfiguration. Our detailed results of the ratio of inner andouter flux distributions agree with the commonly observedbehaviour in several tokamaks. Figure 6 shows HEIGHTScalculations of the NSTX power discharge [31]. The initialparticle temperature was not discussed in [31]; however, as wewill show later that this parameter does not significantly affectthe results. The flowing particle power was estimated from theneutral beam injection power PNBI = 4.9 MW [12].
As is shown in figure 6, the heat flux profile on the innerdivertor surface has two peaks. The experimental resultsshown for NSTX indicate that the additional peak appearswhen the shot power is increased. Our modelling results, forthe first time, show similar two-peak formation and to explainthis effect we tracked individual particles that formed thesecond peak. Examples of the escaped core particle trajectoriesare shown in figure 7 with white lines having the highest impactflux. Arrow 2 shows the second peak location. This location isthe arrival spot and concentration of the particles departed fromthe X-point region. Comparison of figures 4 and 7 brings to aconclusion that peak #2 exists originally, i.e. this main source
Figure 7. Poloidal cross-section of the impact particle flux inNSTX: white lines show particle trajectories; arrows 1 and 2 showthe location of heat load peaks at the inner divertor.
Figure 8. Dependence of NSTX divertor heat flux footprints on theescaped core particle initial temperature.
of heat flux on the inner divertor is due to the ‘leakage’ fromthe X-point area. Therefore, we can define the trajectoriesand the drift processes from the tokamak high-field side as#1 and from the X-point area as #2. The direct impact ofprocess #1 is negligible because of particles returning to coreplasma (see trajectories in figure 7). Increasing the dischargepower stimulates and enhances process #1 corresponding tothe experimental data [31].
However, this process cannot be simply explained bychanging the temperature of escaping plasma with increasingdischarge power. We simulated the escaping core plasmaparticles with various initial temperatures in the NSTX andITER magnetic configuration. Figure 8 shows interestingresults that increasing the initial temperature of the escapingparticles tends to decrease the particle flux in trajectories #1.Moreover, it also tends to increase the probability of particlesreturning to the core on the low-field side and to the slightdecrease in the outer divertor flux.
Figure 9 demonstrates an opposite behaviour of thedependence of the escaped core particles on the initialtemperature in a similar analysis but for ITER geometry andmagnetic configuration. An increase in the initial temperatureof the escaped core particles tends to slightly increase thedivertor heat flux.
Both figures 8 and 9 indicate that the initial temperatureof the escaped particles does not directly play a major role inthe magnitude of the divertor heat flux (in the reasonable limitsof used temperature values). The magnetic field configurationand the generated radial electric field are the main parameters
6
Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
Figure 9. Dependence of ITER outer divertor heat flux on the initialtemperature of the escaped core particles.
Figure 10. Input of electrons and ions energy to the heat load onNSTX divertor surface.
that determine core particle input into the SOL area. The effectof the radial electric field on the escaping core particles iscurrently under investigation.
The drift of the charged particles due to the gradientof the magnetic field has an opposite direction for ions andelectrons. The large mass difference between these particlesand the resulting Larmour radii inequality makes the escapingprobability of core plasma different for electrons and forions. The radial electric field appears due to charged particleseparation around the LCFS. The generated electric fieldcannot totally compensate for the charged particle separationbecause of the presence of the magnetic field. This is clearfrom the analysis of equations (3) and (4) where the last rightterm of these equations has only E/B expressions and does notinclude q in the poloidal plane. Exception is given only for thez′ direction that was chosen to be parallel to the local magneticfield, i.e. compensation is effective only in the tokamak toroidaldirection. In the poloidal direction, the charged particles candrift uncompensated and the final divertor heat distribution iscomposed of the separate flows of electrons and ions [32]. Thisresults in the shaping of the energy deposition profile along thedivertor surface and in depth because of the deeper electronpenetration and the biasing in the SOL and on PFCs [33, 34].Figure 10 shows the spatial distribution of the heat flux onNSTX divertor surface where the input of the electrons andions are shown separately. As is shown, the inner divertor heatload is formed mainly due to energy deposition of the ions.
The flux shapes presented above are for the case of nodivertor plasma. These results are of obvious importance
Figure 11. Contributions of electron and ion energy to heat load onthe ITER divertor surface.
as initial conditions for modelling edge plasma phenomenaand for PFC damage and possible mitigation techniques.Information about separate contributions of the divertor heatflux due to ions and electrons is very limited in spite on thefact that electrons and ions play very different roles in theundersurface processes. Figure 11 shows prediction of theheat load composition for the ITER divertor surfaces. Asshown previously in figure 10, the electron flux is concentratedat the strike points but opposite to the NSTX case, i.e. theelectron part is much larger in the ITER case. Our simulations(figures 8 and 9) indicated the weak dependence of the heatflux spatial profile on the initial particle temperature and we arecurrently analysing the heat flux dependence on the magneticfield configuration.
5. Conclusions
We have developed multidimensional comprehensive modelsfor extensive and integrated simulation of the escaped coreplasma impact on the tokamak divertor and nearby locations.The 3D kinetic Monte Carlo model of the escaping coreparticles is developed for the self-consistent analysis of theMHD evolution of the initiated edge plasma where theescaping particles are used as the input volume source. Theescaped plasma part simulates particle evolution (motion andscatterings) in full 3D in the presence of local magnetic andelectric fields starting from the LCFS through the entire SOLarea and up to penetration inside tokamak components. Thequadtree adaptive mesh refinement algorithms with 5 levelsublayers were developed and implemented in HEIGHTS tosignificantly enhance the accuracy of the calculation and toreduce the computational time. We implemented, verifiedand benchmarked our new Monte Carlo kinetic model in ourHEIGHTS simulation package to study the spatial profile ofion and electron energy deposition of the escaped core particlesof both inner and outer divertor plates for NSTX and ITERdevice parameters, magnetic field complex structure and thecomponents’ geometry. We also studied the influence of initialtemperature of the core escaping particle on the divertor heatload footprints. For accurate prediction of the actual divertordamage and to gain more physical insights of various processeswe calculated, for the first time, details of the electron–ioncomposition and contribution to the surface heat fluxes at boththe inner and outer divertor plates. The distribution profile
7
Nucl. Fusion 53 (2013) 073023 V. Sizyuk and A. Hassanein
was in good agreement with available data and explained thedifferent contributions and sources among ions and electronsand their dependences on machine parameters.
Acknowledgment
This work is supplied by the US Department of Energy, Officeof Fusion Energy Sciences.
References
[1] Kulsrud R.M. 1983 MHD description of plasma Handbook ofPlasma Physics vol 1 Basic Plasma Physics edM.N. Rosenbluth and R.Z. Sagdeev (Amsterdam:North-Holland)
[2] Hu Q. et al 2012 Nucl. Fusion 52 083011[3] Lore J.D. et al 2012 Nucl. Fusion 52 054012[4] Militello F. et al 2012 Plasma Phys. Control. Fusion
54 095011[5] Paccagnella R. et al 2009 Nucl. Fusion 49 035003[6] Kugel H.W. et al 2011 J. Nucl. Mater. 415 S400[7] Braginskii S.I. 1965 Transport processes in a plasma Reviews
of Plasma Physics vol 1, ed M.A. Leontovich (New York:Consultants Bureau)
[8] Erents S.K. et al 2007 J. Nucl. Mater. 363–365 565[9] Loarte A. et al 2007 Nucl. Fusion 47 S203
[10] Shaing K.C. 1997 Phys. Plasmas 4 3320[11] Angioni C. et al 2009 Plasma Phys. Control. Fusion
51 124017
[12] Stotler D.P. et al 2010 Contrib. Plasma Phys. 50 368[13] Todo Y. et al 2012 Nucl. Fusion 52 033003[14] Budny R.V. et al 2012 Nucl. Fusion 52 023023[15] Frerichs H. et al 2012 Nucl. Fusion 52 023001[16] Kritz A.H. et al 2011 Nucl. Fusion 51 123009[17] Coenen J.W. et al 2011 Nucl. Fusion 51 083008[18] Sizyuk V. and Hassanein A. 2010 Nucl. Fusion
50 115004[19] Gray T.K. et al 2011 J. Nucl. Mater. 415 S360[20] Kreter A. et al 2009 J. Nucl. Mater. 390–391 38[21] Sizyuk V. and Hassanein A. 2009 Nucl. Fusion 49 095003[22] Hassanein A., Morozov V., Tolkach V., Sizyuk V. and
Konkashbaev I. 2003 Fusion Eng. Des. 69 781[23] Hassanein A. and Konkashbaev I. 2003 J. Nucl. Mater.
313–316 664[24] Samtaney R. et al 2004 Comput. Phys. Commun. 164 220[25] Lee W., Borthwick A. and Taylor P., 2011 J. Comput. Phys.
230 4848[26] Rognlien T.D. et al 2007 J. Nucl. Mater. 363–365 658[27] Boyd T.J.M. and Sanderson J.J. 2003 The Physics of Plasmas
(Cambridge: Cambridge University Press) p 33[28] Krey U. and Owen A. 2007 Basic Theoretical Physics (Berlin:
Springer) p 81[29] Sizyuk V., Hassanein A. and Bakshi V. 2007
J. Micro/Nanolith. MEMS MOEMS 6 043003[30] Hassanein A., Sizyuk V., Sizyuk T. and Morozov V. 2007
Proc. SPIE 6517 65171X[31] Mastrovito D. et al 2003 Rev. Sci. Instrum. 74 5090[32] Federici G. et al 2003 J. Nucl. Mater. 313–316 11[33] Groth M. et al 2009 Nucl. Fusion 49 115002[34] Zweben S.J. et al 2009 J. Nucl. Mater. 390–391 417
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