Accelerated procedure to solve kinetic equation for neutral atoms in a hot plasma

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1 Journa of Physics: Conference Series PAPER OPEN ACCESS Acceerated procedure to sove kinetic equation for neutra atoms in a hot pasma To cite this artice: Mikhai Z. Tokar 217 J. Phys.: Conf. Ser View the artice onine for updates and enhancements. This content was downoaded from IP address on 29/1/218 at 8:4

2 Acceerated procedure to sove kinetic equation for neutra atoms in a hot pasma Mikhai Z. Tokar Forschungszentrum Jüich GmbH, Institut für Energie- und Kimaforschung - Pasmaphysik, Jü, Germany E-mai: m.tokar@fz-jueich.de Abstract. The recombination of pasma charged components, eectrons and ions of hydrogen isotopes, on the wa of a fusion reactor is a source of neutra moecues and atoms, recycing back into the pasma voume. Here neutra species participate, in particuar, in charge-exchange c-x) coisions with the pasma ions and, as a resut, atoms of high energies with chaoticay directed veocities are generated. Some fraction of these hot atoms hit the wa. Statistica Monte Caro methods normay used to mode c-x atoms are too time consuming for reasonaby sma eve of accident errors and extensive parameter studies are probematic. By appying pass method to evauate integras from functions, incuding the ion veocity distribution, an iteration approach to sove one-dimensiona kinetic equation [1], being aternative to Monte Caro procedure, has been tremendousy acceerated, at east by a factor of 3-5 [2]. Here this approach is deveoped further to sove the 2-D kinetic equation, appied to mode the transport of c-x atoms in the vicinity of an opening in the wa, e.g., the entrance of the duct guiding to a diagnostic instaation. This is necessary to determine firmy the energy spectrum of c-x atoms penetrating into the duct and to assess the erosion of the instaation there. The resuts of kinetic modeing are compared with those obtained with the diffusion description for c-x atoms, being stricty reevant under pasma conditions of ow temperature and high density, where the mean free path ength between c-x coisions is much smaer than that ti the atom ionization by eectrons. It is demonstrated that the previous cacuations [3], done with the diffusion approximation for c-x atoms, overestimate the erosion rate of Mo mirrors in a reactor by a factor of 3 compared to the resut of the present kinetic study. 1. Introduction In devices for thermonucear fusion of hydrogen isotopes deuterium and tritium these are in the form of a hot strongy ionized pasma. To avoid too strong interactions of this pasma with the machine wa there is normay a periphera edge ayer, the so caed scrape-off ayer SOL), where pasma partices and energy is transported aong magnetic fied ines into a specia voume, the so caed divertor [4]. In a future reactor ike DEMO [5, 6] processes in the divertor wi significanty restrain this fow in the main part of the SOL [2] and a significant fraction of pasma partices, diffusing from the pasma core, wi reach the vesse wa before they are exhausted into divertor. Arising pasma fuxes to the wa, with a density of 1 2 m 2 s[2], saturate the wa with fue partices in a time much shorter than the discharge duration. Thus, a comparabe infux of neutra species wi recyce back into the pasma. These neutra partices are not confined by the magnetic fied and can penetrate deep, at severa centimeters into the SOL pasma. Here charge-exchange interactions with pasma ions generate atoms of energies much higher than that Content from this work may be used under the terms of the Creative Commons Attribution 3. icence. Any further distribution of this work must maintain attribution to the authors) and the tite of the work, journa citation and DOI. Pubished under icence by Ltd 1

3 of primary recycing partices. A noticeabe fraction, up to 6%, of such secondary hot neutras escape to the wa, can get into diagnostic ports and destroy instaations there, see figure 1. Figure 1. Processes near an opening in the reactor wa. Statistica Monte Caro methods [7] are normay used to mode c-x atoms at the edge of fusion devices. A crucia obstace to appy these approaches routiney for extensive parameter studies, e.g., with the aim to optimize the duct geometry, is too arge cacuation time for a reasonaby sma eve of accident errors. In a one-dimensiona geometry an aternative approach based on iteration procedure to sove the kinetic equation represented in an integra form has been eaborated in Ref.[1]. Being, unikey to Monte Caro, free from statistica noise and permitting the convergence of iterations to the error eve defined by the machine accuracy, this method, nonetheess, is aso time consuming. However, by appying the pass method to evauate integras from functions, incuding the veocity distribution of pasma ions, a tremendous acceeration of kinetic cacuations, at east by a factor of 3-5, has been achieved in Ref.[2]. This has aowed to perform a thorough comparison of cacuations for 1-D transfer of c-x atoms in a hot pasma in the DEMO SOL done both with the kinetic description and by appying the so caed diffusion approximation. Stricty speaking, the atter is reevant under pasma conditions of ow temperature and high density, e.g., in the divertor, where the mean free path ength between c-x coisions is much smaer than that ti the atom ionization by eectrons. Nonetheess, as it has been demonstrated in Ref.[2], even for the conditions of a hot SOL pasma the resuts of cacuations done with the diffusion approximation deviate ess than by 3% from those obtained kineticay and can be successfuy appied for first estimates. For diverse purposes numerous openings wi be made in the reactor wa, e.g., for ducts eading to optica diagnostic instaations coecting the radiation from the pasma. Such instaations wi be protected from the direct impact of charged pasma partices, moving mosty aong the magnetic ines and penetrating into ducts on a Larmor radius depth of 1mm. Hot c-x atoms, unconfined by the magnetic fied, can, however, freey hit and erode these instaations. Since at the position of an opening the infow of recycing neutras, and, thus, the source of c-x atoms are absent, the transfer of the atter has two- or even three-dimensiona pattern. A 2

4 2-D situation in the vicinity of a circuar opening has been modeed in Ref.[3], by soving a 2-D diffusion equation for c-x atoms. In spite of the reevance of the diffusion approximation in the 1-D geometry, considered in Ref.[2], its appicabiity for muti-dimensiona situations is, however, sti questionabe. In the present paper the approach deveoped in Ref.[2] to acceerate 1-D kinetic cacuations is eaborated further to sove a 2-D kinetic equation for modeing of c-x atoms in the vicinity of a circuar opening in the wa. The resuts of cacuations are compared with those found in Ref. [3] by appying the diffusion approximation for c-x atoms. The remainder of the paper is organized as foows. In the next section the basic equations are formuated and reduced to the form used for the numerica treatment. A pass method appied to asses the integra in the veocity space is aso discussed here. In section 3 the resuts of cacuations both with kinetic and diffusion description of c-x atoms are presented and compared. Concusions are drawn in section Basic equations Henceforth we operate in the coordinate system shown in figure 1 and characterized by the distances x and ρ from the wa and opening axis, respectivey, under the assumption of the opening axis-symmetry. The veocity distribution function of c-x atoms, f x, ρ, V x,v ρ ), is governed by the kinetic equation: V x x f + V ρ ρ ρ ρf) = ν a f + S cx πvth 2 exp V x 2 + Vρ 2 ) Vth 2, 1) where ν a = n kion a + ka cx)is the tota frequency of the atom disintegration in the pasma through the processes of ionization by eectrons and charge-exchange with ions, characterized by the rate coefficients kion a and ka cx, respectivey; n is the pasma density, the same for eectrons and ions due to quasi-neutraity; S cx = Scx +nkcxn a cx is the density for c-x atoms source where the former contribution Scx, specified in Refs. [2, 3], is due to charge-exchange of recycing neutras, and the atter is owing to further c-x interactions of atoms generated from recycing species; the pasma ions with the mass m i and temperature T i have a Maxweian veocity distribution with the therma veocity V th = 2T i /m i Reduction to integra-differentia equations Henceforth, V ρ is discretized as ±Vρ, with Vρ = ΔV 1/2), ΔV = V max / max and = 1,, max. The distribution functions ϕ ± of partices with ρ-veocity component in the range ±Vρ ΔV/2 V ρ ±Vρ +ΔV/2 are governed by the equations foowing from the integration of equation 1) over these ranges: V x x ϕ ± ± V ρ ) ρ ρ ρϕ ± = ν a ϕ ± + S cx, 2 exp V 2 ) x πv th Vth 2, 2) where S cx, = Scx + nkcxn a ) cx δ,with { [ )] [ )]} V ρ +ΔV/2 V ) ρ ΔV/2 Vmax δ = erf erf /erf V th V th V th being the fraction of ions in the the -range. From equations ) 2) one gets equations for the new variabes ϕ = f + + f and γ = Vρ f + f : V x x ϕ + 1 ρ ρ ργ )= ν a ϕ + S cx, exp V 2 ) x πvth Vth 2, 3) 3

5 x γ + ) 2 Vρ V x ρ ρ ρϕ )= ν a V x γ, 4) The atter equation is straightforwardy integrated with respect to x, providing with γ = ) 2 Vρ V x ρ x x ) Uy U x ρ ρϕ )exp dy, U x,y = x,y V x ν a dz, 5) x being the position where the boundary conditions are imposed: since there is no infux of c-x atoms from the wa, γ x =)=forv x > ; besides neutra species are absent deep enough in the pasma, i.e. γ x ) forv x <. By substituting the found expression for γ into equation 3), one gets: V x x ϕ = ν a ϕ + ) 2 Vρ V x ρ x x ρ 2 Uy U x ρϕ )exp V x ) dy + S cx, exp πvth V 2 x V 2 th In the second term on the right hand of the equation above both factor are expanded into the Tayor series up to inear terms, ρ 2 ρϕ )y) ρ 2 ρϕ )x) + x ρ 2 ρϕ )x) y x) and U y U x + ν a x) y x). By taking into account that that the width of the region occupied by neutra atoms is of severa mean free path engths MFPL), i.e. typicay x x > V x /ν a 2 for most x, the term in question is estimated as Vρ) 2 ρ ρϕ ) / ν a ρ). As a resut we get the foowing equation: V x x ϕ = ν a ϕ + ) 2 Vρ ρ 2 ρϕ ) ν a ρ + S cx, exp πvth V 2 x V 2 th ) ). 6) This can be integrated as the first order equation with respect to x, with the same boundary conditionsasforγ, i.e. ϕ x =)=forv x > andϕ x ). Finay, for the density of atoms with Vρ ΔV/2 V ρ Vρ +ΔV/2, η = ϕ dv x, one obtains the foowing integra-differentia equation: where D = V ρ D ν a ρ 2 ρ ρη )= S cx, I xy πvth dy η, 7) ) 2 /νa and I α = F α du with F α u) = u 1 exp u 2 α/u ) and α = U y U x /V th y). The tota density of c-x atoms n cx = m =1 η. Equation 7) is reducing to an integra one obtained in the 1-D case, either equation 16) in Ref.[1] or equation 11) in Ref.[2], by negecting the ρ-derivative and considering one Vρ-vaue ony, i.e. by assuming max = 1. In this case 1, δ =1,n cx = η 1 and S cx,1 = Scx + nkcxn a cx. As it was demonstrated in Ref.[2] this integra equation can be reduced to a ordinary differentia diffusion equation if i) the eectron temperature is ow and the ionization rate is much smaer. 4

6 than that of charge-exchange, kion a ka cx, and ii) the characteristic dimension for the pasma parameter change are much arger than the typica atom MFPL V th /ν a. The same procedure can be done in the case under consideration, providing the 2-D diffusion equation: D th ρ ρ ρ ρ n cx ) x [ ] x n cx T i ) = Scx k ν a m ionnn a cx, 8) i with D th = V 2 th / 2ν a). The boundary conditions to equation 8) impy: i) c-x atoms eave the pasma with the veocity cose to the ion therma one; ii) neutra species are absent far from the wa, n cx x ) =. Additionay, ρ η, ρ n cx =forρ = and ρ for equations 7) and 8) Assessment of the veocity space integra I α By inspecting equation 7) one can see the origin of arge cacuation time for kinetic computations of c-x atoms. Even in the 1-D case for each grid point one has to cacuate encosed doube integras over the ion veocity space and over the whoe pasma voume, y r w, and repeat this in an iterative procedure with respect to the whoe profies of n cx. For the 2-D situation the runs through the grid points in the ρ, V ρ )phase subspace are invoved into cacuations additionay; however these can be efficienty paraeized. Fortunatey, the integra I α, resuting from the generation of c-x atoms with the ion Maxweian distribution function and the consequent attenuation in the ionization and c-x coisions, can be assessed with a high enough accuracy by an approximate pass method [8]. The function F α u) is shown in figure 2a for α =.3, 1 and 3. One can distinguish four u-intervas where F α u) behaves principay differenty: u u 1, u 1 <u u m, u m <u u 2 and u 2 <u, with the points u 1,2 and u m defined by the conditions F α u m )=andf α u 1,2 ) = for the derivatives F α = F α g 1 /u 2, F α = F α g 2 /u 4 where g 1 u) =2u 3 + u α, g 2 u) =g 1 + u) 2 2+6u 2) u 2, u m is a root of a cubic equation, u m = 3 b + a 3 b a with a = α/4 andb = a 2 +1/216. By searching for u 1,2, one can see that for the roots of interest the equation g 2 u 1,2 ) = reduces to the foowing ones: h 1 u 1 )=g 1 u 1 )+u 1 + u 1 2+6u 2 1 =, h 2 u 2 )=g 1 u 2 )+u 2 u 2 2+6u 2 2 =. By seecting propery the initia approximations to u 1,2, these equations are soved with the Newton method, providing accurate enough soutions after 3-5 iterations. To assess the integras I α we approximate function F α u) as a inear one at u u 1 and by poynomias of the fifth order at the intervas u 1 <u u m and u m <u u 2 with the coefficients seected to reproduce F α,m,1,2 = F α u m,1,2 ), F α1,2 = F α u 1,2 ), F αm = F α u m )and F α u m )=F α u 1,2 ) = ; in addition F α u>u 2 ) exp u 2 ) α/u 2 /u2 is assumed. This resuts in: I α α) F α1 2 + F α1δ 1 + F αm u 2 u 1 )+F α2 δ 2 + 9) 2F α1 2 + F α1 δ2 1 F α2 δ2 2 + F δ1 3 + δ3 2 π 1 erfu 2 αm + exp α ), u 2 u2 where δ 1,2 = u m u 1,2. Figure 2b shows I α versus α found with the pass method outined above and evauated numericay. By approximating the direct numerica resuts very accuratey, the procedure above aows to reduce the CPU time by a factor of 3-5 at east. 5

7 As an aternative to the usage of formua 9) one can cacuate in advance the integras I α α) and save this in a tabe. Then, for any particuar α, the integra I α α) can be interpoated from the vaues in the tabe for arguments arger and smaer than that in question. However, the searching of the correct α-interva in this tabe requires aso some time which grows up with α sincei α as E 1 α) and the density of data in the tabe has to be increasing. Thus, it is not obvious that the usage of tabes is more time saving than formua 9). Figure 2. The functions f α u) =u 1 exp u 2 α/u ) for α = 1 soid curve), α =.3 dashed curve) and α = 3 dashed-dotted curve) a); the integra I α α) = F α u) du estimated by the pass method soid curve) and computed numericay dashed curve) b). 3. Resuts of cacuations The radia profies of the pasma parameters used for cacuations of c-x atom parameters are presented in figure 3 and were computed in Ref.[2] with the input parameters from the European DEMO project [5, 6]. In figure 4 we demonstrate the variation of the c-x atom density with the distance from the wa, far from a circuar opening of the radius ρ =.5m and at the opening axis. There is a noticeabe difference between the profies cacuated kineticay and in the diffusion approximation. To comprehend the importance of this difference we assess the rate of erosion by c-x atoms of a Mo mirror imposed in the duct at the distance h from its opening. In a strict diffusion approximation the energy distribution function of c-x atoms coincides in any point in the pasma with that of the ions. Thus, the averaged energy of atoms entering the opening in the wa is cose to the ion temperature at this position, T i ). However, the ion temperature increases by severa times at the penetration depth of c-x atoms, compare figures 3 and 4. C-x atoms generated here party escape without further coisions with eectrons and ions. Therefore, the distribution over the energies E of atoms, hitting the opening, can significanty differ from the oca Maxweian one of magnetized ions: 2 [ E ϕ i x =,E)= πt 1.5 i ) exp E ]. 1) T i ) 6

8 Figure 3. The profies of the pasma density soid curve), the ion dashed curve) and eectron dashed-dotted curve) temperatures used for cacuation of c-x atoms. Consider an infinitesimay thin ring of the width dρ, thickness dx, with the radius ρ and situated in the pasma at the distance x from the wa. C-x atoms of the energy within the range E,E+dE are generated in this ring at the rate: dr cx x, ρ, E) =S cx x, ρ) ϕ i x, E)2πρdρdxdE. 11) Since c-x atoms move in a directions some of them hit the mirror inside the duct, see figure 1. Henceforth we consider the surface position at the duct axis ony. A simpe geometric anaysis shows that c-x atoms from the rings with ρ ρ max = ρ 1 + x/h) can hit this surface point. The contribution of the pasma voume in question to the density of the c-x atom fux perpendicuar to the instaation surface is: s exp λ/s) dj cx x, ρ, E) =dr cx ], 12) 4π [x + h) 2 + ρ 2 [ where s = 1+ρ 2 / x + h) 2] 1/2 is the cosine of the atom incidence ange with respect to the port axis. The exponentia factor in equation 12), with λ = x ν adx / 2E/m, takes into account the destruction, due to ionization and charge-exchange coisions, of the atoms in question on their way through the pasma. By ionization the atom competey disappears. By charge-exchange a new atom is generated. The atter process is taken into account by the second term in the source density, S cx = Scx + nkcxn a cx, and by the summation/integration of dj cx contributions from rings situated in the pasma at a possibe x. The energy spectrum of atoms hitting the mirror is characterized by their fux density in the energy range de, γ = dj cx /de, where the integration is performed over the ring radius ρ and distance from the wa x. By proceeding from ρ to s, accordingtothereation 7

9 Figure 4. The profies of the c-x atom densities far from the opening soid curve) and at its axis dashed curve) computed with diffusion approximation a) and kineticay b). ρ =x + h) 1/s 2 1, one gets: γ h, ρ,e)= r w ϕ i x, E) dx 1 h/ h 2 +ρ 2 S cx 2 exp λ ) ds. 13) s The density of the outfow of the mirror materia partices eroded by physica sputtering with c-x fue atoms can be assessed as foows: Γ sp h, ρ )= de r w ϕ i x, E) dx 1 h/ h 2 +ρ 2 S cx 2 exp λ ) Y sp E,s) ds 14) s Here Y sp is the sputtering yied, whose dependence on the projectie energy E and the cosine s of the incidence ange is cacuated by appying semi-empirica formuas from Ref. [1]. The erosion rate h sp =Γ sp /n sp,withn sp being the partice density of a mirror of Mo and measured in nm/year, is demonstrated in figure 5 as a function of the instaation surface position h inside the port, for n cx x, ρ) invoved into S cx cacuated both kineticay from equations 7) and in the diffusion approximation, by integrating equation 8). One can see that the sputtering 8

10 rate found with kineticay assessed n cx x, ρ) is noticeaby beow that determined by using the diffusive approximation for c-x atoms in the pasma. Figure 5. The erosion rate of a Mo mirror versus the surface mirror position inside the duct computed kineticay soid curve) and with diffusion approximation dashed curve). 4. Concusions The iteration approach eaborated in Ref.[1] to sove 1-D kinetic equation for c-x atoms has been deveoped further to describe the transport of these species in 2-D situation, in particuar, in the vicinity of a circuar opening in the wa of a fusion reactor. Unike to Monte Caro methods the iteration one does not generate statistica errors. To perform a thorough comparison with the resuts obtained by using a diffusion approximation for c-x atoms [3], the soving procedure is acceerated by a factor of 5, by appying pass methods to assess integras in the veocity space [2]. Cacuations done by both approaches demonstrate that the diffusion approximation provides significanty overestimated estimates for the erosion rate of a first mirror in a fusion reactor ike DEMO compared to the kinetic cacuations. Thus, in spite of the reevance of the diffusion approach for c-x atoms in 1-D cacuations of the pasma parameters in the SOL, see Ref. [2], the kinetic one seems to be unavoidabe for cacuations of characteristics being very sensitive to the c-x atom energy ike the erosion rate of the reactor wa and diagnostic instaations. References [1] Rehker S and Wobig H Pasma Phys [2] Tokar M Z 216 Phys. Pasmas [3] Tokar M Z, Beckers M, Bie W Erosion of instaations in ports of a fusion reactor by hot fue atoms in press Nuc. Mater. Energy [4] Stangeby P C 2 The Pasma Boundary of Magnetic Fusion Devices Bristo and Phiadephia: Institute of Physics Pubishing) [5] Federici G, Kemp R, Ward D, Bachmann C, Franke T, Gonzaez S, Lowry C, Gadomska M, Harman J, Meszaros B, Morock C, Romanei F and Wenninger R 214 Fusion Eng. Des [6] Wenninger R, Arbeiter F, Aubert J, Aho-Mantia L, Abanese R, Ambrosino R, Angioni C, Artaud J-F, Bernert M, Fabe E, Fasoi A, Federici G, Garcia J, Giruzzi G, Jenko F, Maget P, Mattei M, Mavigia F, 9

11 Poi E, Ramogida G, Reux C, Schneider M, Siegin B, Vione F, Wischmeier M and Zohm H 215 Nuc. Fusion [7] Landau D P and Binder K 215 A Guide to Monte Caro Simuations in Statistica Physics Cambridge: University Printing House) [8] Mathews J and Waker R L 197 Mathematica Methods of Physics 2nd ed. Meno Park: Benjamin). [9] Janev R K, Langer W D, Evans K J, and Post D E J 1987 Eementary processes in hydrogen-heium pasmas Hamburg: Springer) [1] Eckstein W, Garca-Rosaes C, Roth J and Ottenberger W 1993 Sputtering data Report IPP 9/82 Garching: Institute of pasma Physics) 1