On Dust Particle Dynamics in Tokamak Edge Plasma

On Dust Particle Dynamics in Tokamak Edge Plasma Sergei Krasheninnikov University of California, San Diego, USA With contributions from Y. Tomita 1, R. D. Smirnov, and R. K. Janev 1 1 National Institute for Fusion Science, Toki, Gifu 509-59, Japan The Graduate University for Advanced Studies, Toki, Gifu 509-59, Japan 15-17 December, 003, Kyoto, Japan

Outline I. Introduction II. Dust power and particle balance in tokamak III. Dust in magnetized sheath IV. Impact of surface roughness and dust flights V. Dust and core contamination VI. Conclusions

I. Introduction Significant amount of dust particles is observed in the chambers of fusion devices Winter PPCF 1998 But so far an impact of dust on the performance of current fusion devices is not clear

1) evidence of dust particles in fusion devices a) JIPPT-IIU (NIFS, Japan) [ K. Narihara, et al., Nuclear Fusion 37 (1997) 1177 ] _large scattering signal in Thomson scattering system _most cases: after discharge with disruption _ several cases : during discharges m : small solid particles such as flakes and microparticles _ ~ 100 ms after disruption FIG. 1. 1. (a) (a) Raw scattering signals on on the the fift polychromator. X, X, Y and Z denote the the spec channels. Large scattering signals appeared 350 350 ms. ms. Similar signals appeared on on channe 1, 1,,, 5 5 and and 7 7 at at different times. A plasm current Ip Ip is is shown for for reference.

[ J. Sharpe, A. Sagara, et al., J. Nucl. Mater. 313-316 (003) 455] C Au Ti Ti Ti Fe 10 µ m Carbon, steel, and titanium particles collected from LHD coil armor. The gold results from the coating of particles which is used to avoid charging effects on the non-conductive filter substrate.

Sharpe et al., J. Nucl. Mater 003 However, an existence of dust particles in burning plasma experiments poses a high threat: it can contain toxic and radioactive materials and retain tritium

We show that dust in fusion devices can be very mobile and during one shot dust particles can move through edge plasma on distances much larger that the scrape-of-layer (SOL) width and comparable to tokamak radii Our estimates indicate that it is plausible that transport of dust particles can be an important mechanism of impurity transport into core plasma

II. Dust power and particle balance in tokamak Can dust particle survive in fusion plasma? Consider power and mass balance of carbon dust particles accounting for BB radiation, evaporation, RES, and physical and chemical sputtering In order to simplify our estimates we will assume that electron and ion temperatures are close to Ti Te = T~10 ev and plasma density is about 3 10 13 cm 3. Such parameters are rather typical for tokamak divertor plasma

10 1 heating power cooling power P, W/m 10 10 10 8 10 6 n=10 14 cm -3 total n=10 13 cm -3 thermal evaporation n=10 1 cm -3 radiation 10 4 10 1000 000 3000 4000 5000 T d, K

drd /dt, µm/s 10 5 10 3 10 1 10-1 total rate thermal evaporation physical sputtering RES chemical sputtering 10-3 10-5 1000 000 3000 4000 5000 T d, K

10 5 drd /dt, µm/s 10 3 10 1 10-1 n=10 14 cm -3 n=10 13 cm -3 n=10 1 cm -3 10-3 10-5 1000 000 3000 4000 5000 T d, K Yes, in fusion edge plasma dust particle of ~1 µm scale can live long enough to contribute to dust particle transport!

III. Dust in magnetized sheath Dust charge Z d is determined by the ambipolarity of plasma flux to dust particle e Z r d d =Λ T, where Λ ~3 The forces acting on dust particle M d V d dt d = F F E F ( ) = ezde, F = ς πr M nv V V, g = g, F Md fric F d i i p d 3 M = πrdεm B B 4πR sat tor R R,

Estimates show that electric and friction forces dominate in edge plasmas In fusion devices dust, probably, is formed at the wall surface Therefore, plasma environment near the wall (sheath) important for dust dynamics y y plasma flow F fric -ϕ (y) F E B α z x V diam λ d ρ i x λ d E ρ i y

Dust motion in y-direction can be described by effective potential rt Ud( y) = αςfnshtπrd y+ Λ d ϕ( y) e y min y U d (y) In case of small oscillations in the vicinity of y min, such oscillations can be described by the following equation d d y dt dy = Ω d( yd y min ) ν d V, dt Ω 1 d = M d d Ud, dy y= y min 4 1 1 νv = ςfmn i shviπrd, Ω d ~ 10 s >> ν V ~ 1s

In both z- and x- directions friction forces continuously accelerate dust particle Due to this acceleration dust particle gains speed ~ 3 10 3 cm / s in ~10 3 s Therefore, surface inhomogeneity (corner, step) can cause particle to fly V d dust particle V d dust particle wall wall

IV. Impact of surface roughness and dust flights Even less dramatic surface roughness can be a reason for dust particle flights h w λ w particle trajectory wall wall a) Small amplitude of surface wave h w << ρ i d y dt d ( ) = Ω y ymin ( x ), y ( x ) = y + h sin( k x ), min min w w

( x) ( fric x ) d x F = fric x = F dt M M d d t, x res d d w fric ( Resonance occurs at t Ω M k F ) and it is meaningful resonance if Sres tresωd >> 1 For S res >>1 we can integrate the equations of motion and find 1 / π S y d( t) hw Sres sin res ( ) sin dt + π π Ω 4 4 For y () t > ρ h > ρ / S we expect large excursion of dust trajectory d i w i res

b) Large amplitude of surface wave h w >> ρ i In order to overcome the effect of centrifugal force and confine dust particle in effective potential well within sheath it is necessary to obey the inequality ( x) MV d d Ffric ( y) ( x) = < F F Rw R fric α fric, w where Rw 1/ hwkw is the effective wall curvature radius and is distance along the wall passed by the particle Particle cannot be confined anymore within the sheath after 1 α conf h k w w

Analyzing resonance conditions we find that for h w >> ρ i dust particle looses confinement within the sheath before reaching resonance conditions Once particle looses confinement within the sheath it starts to fly and bounce time to time the sheath article trajectory wall

We notice that the rejection of the particle by the wall can be due to both of the sheath electric field and wall structure itself Just to illustrate the process of particle flights, we consider the case where the wall surface is corrugated in x-direction according to the following expression i x yw( x) = ( 1) hw 1 i, for i λ w x i+ 1, (36) λ w

where i is the integer number, and h w and λ w are the effective magnitude and wave-length of the corrugation We will assume that λ β >> h >> ρ and introduce the angle w w i 4h / λ << 1, so that the inclination of the surface to x-coordinate w w w is ±β w We will assume that the energy of dust particle is large and after it penetrates through the sheath it experiences specula reflection by the wall.

Assume that initially dust particle in the sheath was about the minimum of potential well U d located in the sheath ( sh) Then it will be accelerated along the surface by the force F fric until it reaches breaking point of the surface at x than time the energy sh F fric =λ i w / and flies, gaining, by E1 ~ ( ) λ w. (37) With increasing particle energy its re-entry of the sheath occurs at a distance from original take off, which is much larger than λ w

Therefore we can assume that particle randomly hits positive and negative slop of the wall As a result, the pitch angle of the particles leaving the sheath µ dw dw x y V = sign( V ( ) ( ) ) dw xy Vdw (, ), (38) ( ( xy, ) ) ( xy, ) ( y = ( ) ) + ( dw ( x) ) (where Vdw Edw / Md Vdw V and where V dw the particle velocities taken at the moment when particle leaves the (...) are

wall) is randomly changed by β w <<1 and the evolution of µ dw can be described as a diffusive process We will consider only this evolution assuming that averaged value of pitch angle, 0< µ dw < 1 Notice that in this case particle scattering from the wall cannot flip the ( x) sign of V dw which, then, coincides with the direction of diamagnetic ion flow in the sheath, which accelerates dust particle Then µ dw can be found from diffusive equation

d µ dw dt ~ D, (39) µ with the effective diffusion coefficient D µ ~ β t w coll, (40) and where t coll is the time between two consecutive collisions Taking into account that t coll is determined by reversing the sing of y- component of the velocity by friction force we find

t coll ~ ( ( xy, Edw M ) d µ αf sh fric 1 / dw. (41) Next we consider the particle energy gain sh Dust particle is exposed by accelerating force of diamagnetic flow F fric for the time, τ coll, when particle moves through the sheath with the thickness ~ ρ i Therefore, for µ dw > β, we have w

τ coll ~ ρ i ( ( xy, 1 / ) dw E M d µ dw, (4) and de ( xy, sh ( xy, / dw Ffricτ ( 1 coll Edw M ) d dt Assuming E ~ ( xy, dw t coll > E 1 from Eq. (38)-(4) we find. (43) E ( xy, dw sh sh t F ~ Ffricρ α i 8 β M ρ w fric d i 17 /, (44)

µ dw ~ αβ 3 w sh t F M ρ fric d i 1 / / 7. (45) Taking into account that our analysis is limited by µ dw <1, we find that our approximation works for t < t = µ 1 αβ 3 w M F ρ d i sh fric 1 /. (46) By the time t ~ t µ dust particle moves at the distance, L µ ~ ρi 4 αβ, (47) w

along the wall in x-direction and gains the energy ( xy, ( F Edw Md Vd x ) ( y) ~ ( ) ~ M ( d V ) fricρi d ~ ~ αβ µ β In order to satisfy inequality E restriction h w 1 4 < ( ) w i 1 ( xy, dw sh w sh wffric L > E 1 we find the following λ ρ /. (49) µ. (48) At t > t µ x-component of particle velocity can turn to be negative due to scattering off the wall

As a result particles can fly back toward their initial position and start to lose the energy In the case where wavy wall structure like (36) would be in z-direction then treatment of dust particle motion similar to that we performed gives us the following asymptotic dependences ( yz, ) ( sh) dw fric E F t M ~ ( ) 13 / ( yz, ) dw w Edw E1 µ ~ αβ ln d, (50) ( ) ( ), (51) where the notation is obvious

Notice that in this case energy of the particle increases t due to ion friction force caused by plasma flow along the magnetic field lines, which exist in entire recycling region

V. Dust and core contamination Such flights of dust particles can result in their motion toward core and contamination of core plasma with impurity Assuming that main material dust consist of is carbon we find that in each dust particle with r d 4 ~3 10 cm and ~ gcm / ρ d 3 there are N C ~10 13 carbon atoms

In order to maintain impurity fraction, ξ imp, in the core, the carbon influx through separatrix should be about Γ ( sep) ( sep) imp imp H ( sep Γ ) H is the flux of hydrogenic species though separatrix =ξ Γ, where For ξ imp ~10 and Γ H sep ( ) ~10 1 1 s we find Γ imp ( sep) ~10 19 1 s and it would take the flux, Γ d sep required impurity influx ( ) ~10 6 1 s, of dust particle to establish

Assuming that dust particles penetrate into the core through an area ~3 10 4 cm and dust particle speed is ~ 10 3 cm / s we find dust particle density at separatrix nd ~3 10 cm 3

Where this dust may come from? Assume that carbon dust is originated at the walls of main chamber due to plasma flux to the wall However, since carbon sputtering yield, Y C, is about a few percent, in order to maintain about one percent of core impurity fraction, supplied to the core as a dust, we should assume that roughly ~100% of sputtered carbon is transformed into dust and then flies to the core ( sep ) sep sep imp imp H ( ) C H ( ) Γ ~ ξ Γ ~ Y Γ It looks to be unlikely the case!

Another possibility, dust formation in the vicinity of divertor striking points, looks much more plausible! Indeed, due to strong plasma recycling in the divertor, plasma flux ( div) to divertor targets, Γ H, can easily be as high as 0 3 1 s Then carbon influx into core Γ imp written as follows Γ ( sep) ~10 19 1 ( sep) ( imp Y C dust flight ΓH div ) = η η s with dust can be where η dust is the fraction of sputtered carbon converted into dust, η flight is the flying fraction of dust, which flies to the core

With, for example, Y C ~3% of sputtering yield, it gives sputtered carbon flux in divertor to be ~ 3 10 1 s 1 If only η dust ~3% of these carbon atoms will be transformed to dust and, then, only η flight ~10% of dust particles will be launched towards the core, the relevant ( sep) impurity influx Γ imp ~10 19 1 s through the saparatrix will be established Γ ( sep) ( ) imp = ~ YC ηdustηflightγh div 10 19 s 1 Major axis Core Divertor plate Dust flights

VI. Conclusions Due to acceleration by plasma flows, dust particles can have a very high speed (~ 10 3 cm / s and even higher) Interactions of dust particles with surface inhomogeneities (including micro-roughness as well as steps, corners, etc.) can cause dust particle flights toward tokamak core It is feasible that dust formation in and transport from divertor region play an important role in core plasma contamination

However, even then, dust particle density around separatrix is ~3 10 cm 3, which makes it difficult to diagnose