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2 Chapte 5 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge Magd Shouc Addtonal nfomaton s avalable at the end of the chapte Intoducton Thee ae numbe of physcal stuatons whee plasmas neutalty beaks down though bounday layes called plasma sheaths, whch ae ethe fee o n contact wth a wall. The plasma sheaths tanston poblems ae at the heat of an ndustal evoluton whose theme s the desgn of matte on the molecula scale.the study of the chage sepaaton at a plasma edge eques geneally the soluton of the knetc equatons of plasmas whch, fo a collsonless plasma, usually educe to the well-known Vlasov equaton. Some examples fo the soluton of the Vlasov equaton fo sheaths tanston poblems have been pesented n Shouc, 008a, 009a. A poblem of nteest s the poblem nvolvng the geneaton of adal electc felds and polodal flows to acheve adal foce balance at a steep densty gadent n the pesence of an extenal magnetc feld. Ths poblem s of geat mpotance n the steep densty gadents pedestal of the hgh confnement mode (H-mode) n tokamaks, snce t lagely affects the edge physcs of the H-mode. In the pesent wok, we shall study the poblem of the geneaton of a chage sepaaton and the assocated electc feld at the edge of a cylndcal plasma column, n the pesence of an extenal magnetc feld dected along the cylnde axs. In pevous publcatons on ths poblem (Shouc et al., 003, 004, 008b, 009b), we have consdeed the case whee the electons wee fozen by the magnetc feld lnes, wth a constant densty pofle whch changes apdly along the gadent ove an on obt sze. Along the gadent the electons bound by the magnetc feld cannot move acoss ths feld to exactly compensate the on chage whch esults fom the fnte ons gyoadus. Ths effect s especally mpotant fo lage values of the ato / De, whee s the ons gyoadus and De s the Debye length. Accuate calculaton of the chage sepaaton s mpotant fo the accuate calculaton of the self-consstent electc feld. Ths eques also an accuate calculaton of the exact on obts usng a knetc equaton. In the pesent wok, we use an Eulean Vlasov code to study the 0 Shouc, lcensee InTech. Ths s an open access chapte dstbuted unde the tems of the Ceatve Commons Attbuton Lcense ( whch pemts unestcted use, dstbuton, and epoducton n any medum, povded the ognal wok s popely cted.

3 80 Numecal Smulaton Fom Theoy to Industy evoluton of the on dstbuton functon fo the poblem of the chage sepaaton at the edge of a cylndcal plasma, n the pesence of an extenal magnetc feld dected along the cylnde axs. Eulean codes have the advantage of a vey low nose level and make t possble to measue accuately a vey small chage sepaaton (Shouc et al., 003, Shouc et al., 004 & Shouc, 009a,b), and allows accuate esults n the low densty egons of the phase-space. It was ponted out n the analyss of H-mode powe theshold n a tokamak by Goebne et al., 00, that the changes n the electon densty ne and ne n the tanston to H-mode ae small, and changes n Te ae baely peceptble n the data. Electons and ons have a densty pofle whch vaes apdly along the gadent ove an on obt sze. In pevous publcatons (Shouc et al., 003, 004, Shouc, 00, 008a,b, 009a,b) the electons densty and tempeatue pofles wee kept constant. In the pesent wok we wll allow the electons to move, and t wll be suffcent fo the pupose of ou study to descbe the moton of the electons, havng a small gyoadus, by a gudng cente equaton (Shouc et al., 997). Ths allows a moe accuate descpton of the contbuton of the electons, wth espect to the appoxmaton pevously used n Shouc et al., 003, 004, 009a, whee the pofle of the electons was assumed constant n tme. The electons moton acoss the magnetc feld n the gadent egon s lmted by the gudng cente equaton, and the magnetzed electons cannot move suffcently acoss the magnetc feld to compensate the chage sepaaton whch esults fom the ons moton due to the fnte on obts. To detemne ths chage sepaaton at the plasma edge along the gadent, t s mpotant to calculate the on obts accuately by usng an Eulean Vlasov code. The lage the ons gyoadus, the bgge the chage sepaaton and the self-consstent electc feld at the edge. (Hence the mpotant ole played by even small factons of mputy ons, because of the lage gyoad, n enhancng the electc feld at the plasma edge). In full toodal geomety, thee ae neoclasscal effects whch can play a ole n ths poblem, such as the neoclasscal enhancement of the classcal on polazaton dft, o the neoclasscal dampng of polodal flows (Stx, 973, Hshman 978, Waltz et. al., 999). We focus fo smplcty n the pesent wok on a cylndcal geomety fo the poblem of the geneaton of an electc feld and polodal flow at a plasma edge due to the fnte ons gyoadus, when the extenal magnetc feld s appled along the axs of the cylnde. The soluton we pesent s a two-dmensonal (D) soluton n a cylndcal geomety, wth the extenal magnetc feld assumed unfom along ts axs. We compae the adal electc feld calculated along the gadent wth the macoscopc values calculated fom the knetc code fo the gadent of the on pessue n the adal decton, and we fnd that ths quantty balances the adal electc feld faly well, a esult smla to what has been pesented n one-dmensonal geomety (Shouc, 00, Shouc et al. 003, 004). The contbuton of the Loentz foce tem along the gadent s neglgble. Fo the paametes we use n the pesent wok, the soluton allows fo a small value of E to exst, especally on the hgh feld sde of the cylnde as t wll be dscussed below, whch causes a small oscllaton n the adal decton.

4 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 8 As mentoned above, the ons ae descbed by a knetc Vlasov equaton and the electons ae descbed by a gudng cente equaton. These equatons ae solved numecally usng a method of chaactestcs. Thee have been mpotant advances n the last few decades n the doman of the numecal soluton of hypebolc type patal dffeental equatons usng the method of chaactestcs. The applcaton of the method of chaactestcs fo the numecal soluton of the knetc equatons of plasmas and fo the gudng cente equatons has been ecently dscussed n seveal publcaton (see fo nstance Shouc, 008a,c, 009a). These methods ae Eulean methods whch use a computatonal mesh to dscetze the equatons on a fxed gd, and have been successfully appled to dffeent mpotant poblems n plasma physcs nvolvng knetc equatons, such as lase-plasma nteacton (Ghzzo et al., 990, 99, Stozz et al., 006, Shouc, 008d), the calculaton of an electc feld at a plasma edge (Shouc, 00, 008b, 009b, Shouc et al., 000, 003, 004), the applcatons of gyo-knetc codes to study edge physcs poblems n plasmas (Manfed et al., 996, Shouc, 00, Shouc et al., 005, Pohn & Shouc, 008) and to collsonal plasmas (Batshchev et al., 999). These methods pesent the geat advantage of havng a low nose level, and allow accuate esults n the low densty egons of the phase-space (Shouc, 008c). In the applcatons pesented (Shouc, 008a,c, 009a), the computaton was usually done on a fxed gd, so no dynamcal gd adustment was necessay, and ntepolaton was estcted to the use of a cubc splne, whch compaed favouably wth othe methods (see, fo nstance, Pohn et al., 005), so altogethe the method was accuate and emaned elatvely smple. Intepolaton n seveal dmensons usng a tenso poduct of cubc B-splnes has been also successfully appled (Sonnendücke et al., 999, Shouc, 008a, 009a, 0). The method of chaactestcs has been also successfully appled n fluds to poblems havng shock wave soluton (Shouc & Shouc, 007). Lage Couant-Fedechs-Levy (CFL) computaton paamete s possble, and theefoe the tme-step numecal lmtaton by lage veloctes can be emoved, f the physcs makes t possble. A moe complete study on splnes can be found n the book of Ahlbeg et al., 967, and an mpotant theoetcal study on the method of chaactestcs can be found, fo nstance, n the book of Abbott, 966. Moe applcatons to plasma physcs poblems can be found n the seveal efeences we have cted.. The elevant equatons and the numecal method We consde the cylndcal coodnate system (,, z). The plasma s unfom n the z decton. The adal decton n the cylndcal plasma s nomal to a vessel suface whch s located at = max=. The constant magnetc feld s n the z decton whch epesents the toodal decton, and s the polodal decton. The constant magnetc feld n the z decton s gven by: Be 0 B z cos () whee B0 s the magnetc feld along the axs of the cylnde at = /, and / ma whee ma s the mao adus of the tokamak. We assume that at = max= we have / ma =0.. In ths case we can also wte / 0. /. We consde a deuteum plasma, ma

5 8 Numecal Smulaton Fom Theoy to Industy m / me x836. The ons ae descbed by the nomalzed D Vlasov equaton fo the ons dstbuton functon f (,, v, v, t) : f f v f f v E c t v vv f E c v 0 v () whee fom Eq.() the on cycloton fequency 0 / ( cos ). The electons ae descbed by the nomalzed D gudng-cente equaton: whee the dft velocty Vd Ex B/ B gve the followng equaton: n e t c. Vn 0 d e c. Equaton (3) can be developed n ou D system to (3) n n n cos cos t e e e Vd Vd ne e ne e c0 c0 (4) whee cos Vd E ; c0 V d E cos c0 Ths system s coupled to Posson s equaton: ( ) ( n ); ne E ; E (5) Tme s nomalzed to the nvese plasma fequency, velocty s nomalzed to the acoustc velocty Cs Te / m, and length s nomalzed to the Debye length De Cs / p. Te s the electon tempeatue and m s the on mass. The potental s nomalzed to Te / e, and the electc feld s nomalzed to T e / ( e De), and the densty s nomalzed to the peak ntal cental densty. The on cycloton fequency c as pevously defned s nomalzed to p. The system s solved ove a length L = 75 Debye lengths n font of the vessel suface, wth an ntal on dstbuton functon fo the deuteons ove the doman gven n ou nomalzed unts, by: L, p / v v T e f,, v, v n( ) ; (6) T n () and ne( ) ae the ntal on and electon densty pofles espectvely, wth ne() n() n the ntally neutal system. s the adus of the cylnde as we pevously mentoned. We also use the followng paametes:

6 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 83 T T T ; 0 (7) T / e De e c0 p In the pesent calculatons the paamete 0 / 0. /. Wth the ntal dstbuton c functon fo the ons n Eq. (6), we have T T T spatally constant, the facto T n Eq. (7) n the calculaton of the gyoadus takes nto account the fact that the pependcula T tempeatue s v v v m. Equatons () and (3) ae solved by a method of factonal step (Yanenko, 97), fst appled fo a Vlasov equaton by Cheng & Kno, 976, and Gagné & Shouc, 977, coupled to a method of chaactestcs. Fo the geneal case whee seveal dmensons ae nvolved, the factonal step technque allows the educton of the mult-dmensonal equaton to an equvalent set of educed equatons. To advance Eq. () and Eq.(3) fo one tme-step t, the splttng of the equatons s appled as follows. Step. We solve fo t/ the equatons: p f f v f v 0 t n e n e n e Vd Vd 0 t (8) (9) Equatons (8) and (9) ae solved by ntepolaton along the chaactestcs, to be descbed below. We then solve Posson s equaton n Eq.(5) to calculate the new electc feld. Step. We solve fo t the equatons: f v f vv f Evc E cv 0 t v v ne cos cos ne ne t e e c0 c0 (0) () Agan Eq.(0) s solved by a method of chaactestcs, to be descbed below. Step 3. We epeat Step fo t/, and then solve Eq.(5) to calculate the new electc feld n n E, E. Ths completes a one tme-step cycle t... The soluton fo Step Fo the soluton of Eq. (8), we fst solve the chaactestc equatons:

7 84 Numecal Smulaton Fom Theoy to Industy d d v v ; () dt dt at a gven ( v, v ) n velocty space. The soluton of Eqs () ognatng at ( o, o ) at tme t, and eachng the gd pont (, ) at t t t/ can be wtten as follows, fo a half tmestep t/: o t / ; o ln t/ (3) v Fo v t/ «, the second equaton educes to o t /. Theefoe the soluton of Eq. (8) can be wtten, fo half a tme-step t/, as follows: * v f,, v, v, tt/ f v t/, ln, v, v, t v vt/ (4) The ght hand sde of Eq. (4) s calculated by ntepolaton usng a tenso poduct of cubc B-splnes, n whch s peodc (Shouc et al., 004, Shouc, 008a, 009a). Fo each v and v, we wte fo the ntepolaton functon s (, ): N N s (, ) B() B() (5) 0k0 k k takng nto account that s peodc. B() and Bk( ) ae cubc splnes.the cubc B-splne s defned as: 3 ( xx) x x x 3 h 3 h ( xx) 3 h( xx) 3( xx ) x xx B( x) 3 3 6h h 3 h ( x3 x) 3 h( x3 x) 3 3( x x) x x x 3 ( x x) x x x (6) and B( x) 0 othewse. The gd sze h (ethe o n ou notaton), s assumed unfom. Fo the calculaton of the coeffcents k of the cubc B-splne ntepolaton functon s (, ) fo peodc n Eq.(5) see detals n Shouc et al., 004 and Shouc, 008a, 009a. We then solve Eq.(9) fo a half tme-step t/ along the chaactestcs: d cos d E cos Vd(, ) E ; V (, ) (7) dt dt d c0 c0

8 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 85 Equatons (7) ae solved usng an teatve pocess. We assume that at the tme-step t n/ t n t /, s at the gd pont and s at the gd pont. The followng leapfog scheme can be wtten fo the soluton of Eq.(7): ( t ) ( ) ( ) n /4 /4 (, ) ( t n n n, t V n d V d ) t / (8) ( tn) ( ) ( ) /4 /4 t n t n n n Vd (, ) Vd (, ) t / (9) whee ( t ( n), ( tn)) s the pont whee the chaactestc s ognatng at t n (not necessaly a gd pont). Put: we can ewte Eqs.(8,9) as follows: ( tn) ; ( t n) (0) t V (, ) () d 4 t V d (, ) () 4 whch ae mplct equatons fo (, ), and whch ae solved by teaton. Usually, two 0 0 teatons ae suffcent fo convegence. We stat wth ( 0, 0). We get at the fst teaton: t V d(, ) 4 t E 4 cos (, ) c0 (3) t V d(, ) 4 (, ) cos t E (4) 4 c0 Second teaton: cos( ) t t d(, ) (, ) V E (5) 4 4 c0

9 86 Numecal Smulaton Fom Theoy to Industy t V d (, ) 4 E (, ) cos( ) t 4 c0 (6) The quanttes E (, ) and E (, ) ae calculated by a cubc B-splne ntepolaton, smla to what s descbed n Eq.(5). Fnally to advance Eq.(9) by t /, we calculate: * e e n (,, tt/) n (,, t) (7) whee the ght hand sde of Eq.(7) s calculated agan usng a cubc B-splne ntepolaton as ndcated n Eq.(5)... The soluton fo Step We go to Step and solve fo t Eqs.(0,). We fst solve the equaton: f f vv f E c E cv 0 t v v (8) The electc feld E ou calculaton. emaned small, snce E, and ~ =5000 towads the edge n c ; c dv E v v dv E v v v (9) dt dt Followng the same teatve steps as descbed fo Eq.(7), to an ode Ot of Eq. (9) yelds the followng soluton to Eq.(8):,,,,,,,, the soluton f v v tt f v a v b t (30) Agan the D ntepolaton n Eq. (30) s done usng a tenso poduct of cubc B-splne, as explaned n Eq.(5), and a and b ae calculated wth smla teatve steps as n Eqs.(3-6) and ae gven, to an ode O t, by the expessons: t ( b ) a E cv b t ( b )( a ) b E c v a (3) (3)

10 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 87 whee a t E c ; b t E c The densty n ** (,, t t) s calculated by ntegatng Eq.() as follows: e cos cos n (,, t t) n exp( E te t) ** * e e c0 c0 (33) * e whee n s calculated n Eq.(7). To calculate f and, we then epeat Step fo the ** ** soluton of Eqs. (8,9) fo t / usng f and n e calculated fom Eq.(30) and Eq.(33). The n electc feld s then updated to calculate n E (, ) and E (, ) by solvng Posson equaton. n n n e.3. The soluton of Posson s equaton Equaton (5) s solved by fst Foue tansfomng the equaton n the peodc decton : (, ) () e m (34) m and the esultng equaton s then dscetzed n the adal decton ove a unfom gd followng the method of Kno et al., 980 (also edscussed moe ecently n Couselles et al. 009): m whee ( ) Am m, Bmm, Cmm, ( m, 0 m, m, ) (35) m ( ) 0 m ( ) m ( ) A ; B ; C (36) m,,, m m and, ( ) ; n ne ; n f (,,, ) d d m m Equaton (35) s solved usng a tdagonal algothm. To get the bounday condtons, we assume n the pesent calculaton that the deuteons and electons cuents httng the cylndcal wall suface at ae collected by a floatng potental cylndcal vessel. Theefoe we have the elatons fo the chage collected: whee E (, ) ( J J ) o E (, ) ( J J ) dt t (37) e e 0 t

11 88 Numecal Smulaton Fom Theoy to Industy and V d s defned n Eq.(7). E E followng bounday condton at : J (, ) v f,, v, v, t d d ; J e n e V d (38) fom Eq.(37) s used to obtan fo the potental the (39) E By ntegatng the elaton.e, we get a elaton between the chage collected on the cylndcal wall and the chage appeang n the ntally neutal plasma: E E ( L) E d ; whee d L (40) L whee L s the wdth of the plasma slab, whch extends fom L to at the edge of the cylndcal plasma of adus. s the chage appeang n the system. Fom Eq.(40), we get the followng elaton: E E ( E ( d))/( L) L (4) L We assume that the plasma patcles ae allowed to ente o leave at the bounday at L. So the dffeence between the electc felds at the bounday L and at = must be such that Eq. (40) must be satsfed at evey tme-step n evey decton. In the pesent smulaton, E s calculated fom Eq. (37), whch defnes the devatve of the potental at the ght bounday to be used n the soluton of Eq. (35). We fx the potental to be zeo at the bounday at L and solve Posson equaton n Eq.(35) fo the potental. Then the esultng electc feld at E, calculated at L fom Eq. (5), must satsfy Eq. (4). L The ntal densty pofles at the neutal plasma edge ae gven by: L n ( ) n ( ) 0.5( tanh (( / 5) / 4)) (4) e L The pofles n Eq. (4) stuate the steep gadent to be centeed at a dstance of L / 5 fom the wall of the vessel, whch put the plasma elatvely close to the floatng wall of the vessel. The wall of the vessel wll collect the chage comng fom the plasma, especally due to the lage ons gyoadus. The system s solved fo an edge thckness L 75, and =5000 fo the adus of the cylnde (we expect ths adus to be lage n a tokamak, so n the doman (, ), the vaaton of the quantty / emans vey close to. L We use N=50 gd ponts n space n the adal decton, and 8 gd ponts n the azmuthal decton. 80 gd ponts ae used n each velocty decton. The velocty extema fo the ons veloctes ae 4 Te / T n ou nomalzed unts, wth Te/T = n the pesent smulatons. L

12 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge esults The code was executed fo a suffcently long tme to each a steady state. We noted aound t=500 that the code has ndeed each a steady state. Snce c0 0. / (nomalzed to p ), then the gyopeod / c Ths means that the code has executed moe than 5 gyopeods. Fgue() shows the electc feld at = 0 (full cuve), = / (boken cuve) and = (dashed-dotted cuve), at tme t = 495 (left fgue) and t=500 (ght fgue). At the edge and along the gadent the electc feld s dected towads the cente to the nteo of E the plasma. The electc feld at the floatng vessel wall at = was calculated usng Eq. (37) at 8 ponts ove the ccle. We see fom Fg. () that the electc feld at = 0 s hghe (n absolute value) than the one at =, and we note that the cuves fo 0<-<75 have eached a steady state stable equlbum, wth the chage collected at = emanng constant, whle towads the cente fo 75<-<75, the cuve at = shows a small steady state oscllaton aound zeo. The chage collected on the floatng wall of the cylndcal vessel at = 0, / and ae espectvely , and at t=500. Fo the soluton of Posson equaton, we set the value of the electc feld E at the cylndcal vessel wall exactly equal to the chage collected on the wall accodng to Eq.(37). If the chage collected on the wall s equal to the chage appeang n the system, then accodng to Eq.(4) : E E ( d)) / ; E 0 (43) L L whch s what we see fo the cuves at =0 and / n Fg.(). Indeed the code calculate fo E the chage appeang n the system ( d ))/ the values of and , vey close to the values of and calculated fo pevously mentoned. And the quantty 3 (.x0 and 3.x0 at =0 and / espectvely). L E at =0 and / as we E L was neglgble by an ode of magntude In ou code, we allow at L fo the possblty of plasma to flow acoss the bounday. Note that fo ou pesent set of the paametes the electc feld E calculated emaned neglgble fom =0 to / (see Fg.(6) below), and small aound =. Aound =,the chage appeang n the system s not exactly equal to the chage collected at the cylndcal vessel wall, but shows a small oscllaton due the fact that thee s a small plasma cculaton at L. In ths case the electc feld Eq.(4). At t=500, the value calculated fo the ght hand sde of Eq.(4) s E L s calculated fom.44x0, whle the value calculated by the code s.55x0 (whch s the value we see n the ght fgue n Fg.() fo the = cuve ). The ageement s vey good.

13 90 Numecal Smulaton Fom Theoy to Industy Fgue. Electc feld E at 0 (full cuve), / (boken cuve), and (dashed-dotted cuve), at t=495 (left fgue) and t=500 (ght fgue). Fgue() shows the potental at = 0 (full cuve), = / (boken lne) and = (dasheddotted lne), at tme t = 495 (left fgue) and t=500 (ght fgue). The cuves at = 0 and = / show neglgble oscllaton, whle the cuve at = shows a small oscllaton. Snce as ndcated n Fg.() the E pofles ae essentally constant, especally fo 0<-<75 along the gadent, then the oscllaton of the cuves at = (dashed-dotted lne) s takng place n such a way that the slope of the potental E emans constant fo 0<-<75. Only fo 75<-<75 does the small oscllaton of the cuve at = n Fg.() (dashed-dotted lne) tanslates nto a small oscllaton of the electc feld E.. Fgue(3) gves at tme t = 500 and fo = 0 the electc feld E (full cuve). Also at = 0 the dashed-dotted cuve n Fg.() gves the Loentz foce, whch n ou nomalzed unts s gven by 0. v c / p v /(. 0. cos ), and the boken cuve gves the pessue foce P / n, 0.5 P n T T, wth the followng defnton: T dv dv v v f v v (44), (, ) (,, ),,, n v dv dv v f v v n dv dv f v v,,, ; (, ),,, (45),, n

14 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 9 Fgue. Potental at 0 (full cuve), / (boken cuve) and (dashed-dotted cuve) at t=495 (left ggue) and t=500 (ght fgue). Fgue 3. Plot at 0 of the electc feld E (full cuve), the Loentz foce (0. / ) v /(. 0.( / )cos ) (dashed-dotted. cuve), and the pessue foce P / n (boken cuve). The cuve -n/ s plotted fo efeence ( dashed- 3 dotted cuve). At tme t =500. The P / n tem (boken cuve) shows a vey good ageement along the gadent wth the sold cuve fo E, and the Loentz foce appeas neglgble along the gadent. In a egon of about two gyoad fom the wall (aound 40 Debye lengths fom the wall), we have small egula oscllatons n space (and tme), the accuacy of P / n beng degaded by the dvson wth a vey small value of the densty n appeang close to the vessel suface. To avod ths poblem, we plot n Fg. (4) the quanttes (0. / ) n v /(. 0.( / )cos ), ne, P, at 0 ( note that J n v ). We see that thee s a vey nce ageement fo

15 9 Numecal Smulaton Fom Theoy to Industy the elaton n E P along the gadent (the densty n / 0 s also plotted n Fg.(4) to locate the dffeent pofles wth espect to the gadent). The Loentz foce tem emans neglgble n the gadent egon, and emans vey small n the bulk at 75<-<75. The P / n tem s zeo n the bulk snce n and T ae essentally flat n that egon. We have E seen n ths case that the total chage ( d ))/ appeang n the system and calculated by the code by ntegatng the chage as n Eq. (40) and by calculatng E fom Posson equaton, s essentally equal to the electc feld L E. We note that cuves smla to Fg.(4) can be calculated at dffeent angles, showng ne essentally balanced by the P. Fgue 4. Plot at 0 of ne (sold cuve), (0. / ) n v /(. 0.( / )cos ) (dash-dot cuve), and P (boken-cuve). The cuve n/0 s plotted fo efeence (dashed-3 dotted cuve). At tme t=500. The ntal densty pofles ae gven n Eq.(4), so the ntal chage s zeo. Fg.(5) shows the chage densty n ne at the tme t = 500 and 0 (full cuve), / (boken cuve) and (dashed-dotted cuve). It s nteestng to note that the code was able to mantan the steep densty gadents (see Fg.(4) fo a plot of n ). The stable n tme steep densty pofle changes n space apdly along the gadent ove an on obt sze ( / 0 ), and the elaxaton of the steep gadents dung the smulaton detemnes the chage densty n n e. The chage densty s mpotant along the gadent at the plasma edge. The electons, descbed by the gudng cente equaton gven n Eq.(3) cannot compensate along the gadent the chage sepaaton caused by the fnte on gyoadus, whch esults n the chage densty we see n Fg.(5). De

16 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 93 Fgue 5. Chage densty at 0 (full cuve), / (boken cuve) and (dashed-dotted cuve), at t=500. We have noted n Fg.() the constant value of the E pofles, especally fo 0<-<75 along the gadent, tanslates the constant slope of the potental. Howeve, the potental pofle at =, although keepng essentally the same slope fo 0<-<75, shows an oscllaton n tme. Ths esults n a vaaton wth an electc feld E whch, as we mentoned befoe, s small fo the paametes we ae usng. Fgues (6) pesents ths electc feld E at t=485 and t=495. It shows no oscllaton E n at = 0 (full cuve, essentally equal to zeo), a neglgble oscllaton at = / (boken cuve), and a small oscllaton fo the = (dasheddotted cuve), on the hgh feld sde of the cylnde. The cuve at = s nteestng, t shows the oscllatng cuve fo E havng a constant flat value n the gadent egon, and a lnea vaaton n the nne egon. Fgue 6. Electc feld E at 0 (full cuve), / (boken cuve), and (dash-dot cuve), at t=485 (left fgue) and t=495 (ght fgue).

17 94 Numecal Smulaton Fom Theoy to Industy Fgue (7) gves at t = 500 the tempeatue T as defned n Eq. (43), fo = 0 (full cuve), = / (boken cuve) and = (dashed-dotted cuve). The dashed-3 dotted cuve s fo nt, whch s essentally the same fo all angles. Fgue (8) pesents smla esults fo T as defned n Eq.(43). Note n Fg.(7) and Fg.(8) the dvson by the vey small densty at the edge gves egula oscllatons. Howeve the dashed-3 dotted cuve n Fg.(7) and Fg.(8) s vey smooth, and s fo nt and nt espectvely, whch emoves the poblem of the dvson by the vey small densty at the edge. We pesent n Fg.(9) the total pessue P 0.5 nt T fo = 0 (full cuve), = / (boken cuve) and = (dashed-dotted cuve). We see that these cuves ae essentally dentcal, fo the paametes we ae usng n the pesent smulaton thee s no vaaton n fo the total pessue tem. Fgue 7. Tempeatue T at 0 (full cuve), / (boken cuve) and (dashed-dotted cuve). The dashed -3 dotted cuve s fo nt, whch s essentally the same fo all. Tme t=500. Fgue 8. Tempeatue T at 0 (full cuve), / (boken cuve) and (dash-dot cuve). The dashed-3 dotted cuve s fo nt, whch s essentally the same fo all. Tme t=500.

18 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 95 Fgue 9. Pessue 0.5 P n T T (dashed-dotted cuve), at t= 500. at 0 (full cuve), / (boken cuve) and Fgue(0) pesents the plot of the component of the E B/ B unts s wtten E dft, whch n ou nomalzed 0.( / )cos /0., at = 0 (full cuve), = / (boken cuve) and = (dashed-dotted cuve). The cuves fo 0<-<75 have eached a stable equlbum as dscussed n Fg.() fo E, whle towads the cente fo 75<-<75 the cuves at = shows a small steady state oscllaton aound zeo, as pevously mentoned fo E n the ght fgue n Fg.(). Note n Fg. (0) that ths azmuthal (.e. polodal) E B/ B dft s flat close to the edge, and s below the acoustc speed at = at the edge, and above the acoustc speed at = 0 and = / at the edge (veloctes ae nomalzed to the acoustc speed Cs). We plot n Fg. () n E B / B v, whee the damagnetc dft the total polodal cuent (n the decton) D s vd BnT / neb. We see that the total cuent s essentally zeo,.e. the pofle s adustng tself so that the E B/ B dft and the damagnetc dft ae essentally equal and opposte (n ou unts, the total polodal cuent s ne p 0.( / )cos /0., essentally equal to zeo, showng a small oscllaton aound zeo n Fg.(), at the left bounday). So the E B/ B dft s balanced faly well by the damagnetc dft. Ths s the esult we get f we calculate the polodal cuent: We get ndeed a neglgble value fo J (, ) v f,, v, v, t dv dv (46) J. (note that J n v ). We note that due to the small E feld (see Fg.6), the E B/ B dft has a small oscllatng component E ( 0.( / )cos ) n the adal decton, wth a cuve smla to what s

19 96 Numecal Smulaton Fom Theoy to Industy pesented n Fg.(6). Ths adal oscllaton s neglgble aound 0 and /, and s small on the hgh feld sde aound, as pevously dscussed fo E n Fg.(6). Fgue 0. Plot of E 0.( / )cos / 0. fo 0 (full cuve), / (boken cuve) and (dashed-dotted cuve). At tme t=500. Fgue. Plot of the total polodal cuent ne P 0.( / )cos / 0. fo 0 (full cuve), / (boken cuve) and (dashed-dotted cuve). At tme t= Concluson We have pesented n ths wok the self-consstent knetc soluton fo the poblem of the geneaton of a chage sepaaton and an electc feld at a plasma edge, unde the combned effect of a lage ato of the ons gyoadus to the Debye length / (equal to 0 n the smulaton we have pesented) and a steep densty gadent, when the electons whch ae De

20 Chage Sepaaton and Electc Feld at a Cylndcal Plasma Edge 97 bound to the magnetc feld cannot compensate along the gadent the chage sepaaton due to the fnte ons gyoadus. In the cylndcal geomety consdeed, a fully knetc equaton has been used to descbe the ons, and a gudng cente equaton has been used to descbe the electons bound to the magnetc feld. These equatons have been solved usng the method of chaactestcs (Shouc, 008a,b,c,d, 009a). The numecal method used fo the soluton n cylndcal geomety, based on an ntegaton of the equatons along the chaactestcs coupled to a two-dmensonal ntepolaton (Shouc et al., 004) appled successvely n confguaton space and n velocty space, s poducng accuate esults. The poblem of the fomaton of a chage sepaaton s of geat mpotance n the study of the H-mode physcs n tokamaks. We have consdeed the case whee the gadent n the densty pofles s located n font of a floatng cylndcal vessel. So the chage appeang n the system s essentally equal to the chage collected on the walls of the floatng vessel. The soluton shows n the adal decton that the electc feld along the gadent s balanced by the adal gadent of the pessue, and the total polodal cuent s essentally zeo. The pesent esults whee electons ae descbed by a gudng cente equaton ae close to what has been pevously epoted when assumng the electons statonay and fozen by the magnetc feld, and the code shows that a soluton wth a steep gadent, mantanng a chage sepaaton, whee the electon and on denstes vay apdly ove an on gyoadus, s possble. Also the calculaton wth the pesent set of paametes whch allow a small value of the polodal feld E to exst, shows the pesence of a small oscllaton of E on the hgh feld sde aound (see Fg.(6)), to whch s assocated a small adal oscllaton due to the adal component of the E B/ B dft, equal to E 0.( / )cos /0. n ou nomalzed unts. The pesent code s a step close to a code whch wll nclude neoclasscal effects due to toodal geomety, whch can play a ole n ths poblem, such as the neoclasscal enhancement of the classcal on polazaton dft, o the neoclasscal dampng of polodal flows (Stx, 973, Hshman 978, Waltz et. al., 999). Autho detals Magd Shouc Insttut de echeche Hydo-Québec (IEQ), Vaennes, Québec, Canada Acknowledgement The autho s gateful to D. eean Gad fo hs suppot, and to the Cente de calcul scentfque de l IEQ (CASI) fo compute tme fo the smulatons pesented n ths wok. 5. efeences Abbott, B.A. (966). An Intoducton to the Method of Chaactestcs; Thames and Hudson, London.

21 98 Numecal Smulaton Fom Theoy to Industy Ahlbeg, J.H., Nlson, E.N. & Walsh, J.L. (967). The Theoy of Splnes and the Applcatons; Academc Pess, New Yok, N.Y. Batshchev, O., Shouc, M., Batshcheva, A. & Shkaofsky, I. (999). Fully Knetc Smulaton of Coupled Plasmas and Neutal Patcles n Scape-off Laye Plasmas of Fuson Devces. J. Plasma Physcs, Vol. 6, pp Cheng, C.Z. & Kno, G. (976). The Integaton of the Vlasov Equaton n Confguaton Space. J. Comp. Phys., Vol., pp Couselles, N., espaud, T. & Sonnendücke, E. (009). A Fowad Sem-Lagangan Method fo the Numecal Soluton of the Vlasov Equaton. Comp. Phys. Comm., Vol. 80, pp Gagné,. & Shouc, M. (977). A Splttng Scheme fo the Soluton of a One-Dmensonal Vlasov Equatons. J. Comp. Phys., Vol. 4, pp Ghzzo, A., Betand, P., Shouc, M., Johnston, T., Fex M. & Falkow, E. (990). A Vlasov Code fo the Numecal Smulaton of Stmulated aman Scatteng. J. Comp. Phys., Vol. 90, pp Ghzzo, A., Betand, P., Shouc, M., Johnston, T., Falkow, E., Fex M. & Demchenko, V.V. (99). Study of Lase-Plasma Beat Wave Cuent Dve wth an Eulean Vlasov Code. Nucl. Fuson, Vol. 3, pp Goebne,.J., Thomas, D.M. & Deanan,.D. (00). Evdence fo Edge Gadents as Contol Paametes of the Spontaneous Hgh-Mode Tanston. Phys. Plasmas, Vol. 8, pp Hschman, S.P. (978). The Ambpolaty Paadox n Toodal Dffuson, evsted. Nucl. Fus., Vol.8, pp Kno, G., Joyce, G. & Macus, J. (980). Fouth-Ode Posson Solve fo the Smulaton of Bounded Plasmas. J. Comput. Phys., Vol. 38, pp Manfed, G., Shouc, M., Dendy,.O., Ghzzo, A. & Betand, P. (996). Vlasov Gyoknetc Smulatons of Ion-Tempeatue-Gadent Dven Instabltes. Phys. Plasmas, Vol. 3, pp. 0-7 Pohn, E., Shouc, M. & Kamelande G. (005). Eulean Vlasov Codes. Comp. Phys. Comm., Vol. 66, pp Pohn, E. & Shouc, M. (008). Collsonless Dffuson of Patcles acoss a Magnetc feld at a Plasma Edge n the Pesence of Tubulence. Poc. Vlasova wokshop, n Commun. Nonlnea Sc. Nume. Smul., Vol. 3, pp Shouc, M., Lebas, J., Kno, G., Betand, Ghzzo, A., Manfed, G. & Chstophe, I. (997). Effect of Vscous Dsspaton on the Geneaton of Shea Flow at a Plasma Edge n the Fnte Gyoadus Gudng Cente Appoxmaton. Physca Scpta, Vol. 55, pp Shouc, M., Pohn, E., Kno, G., Betand, P., Kamelande, G., Manfed, G. & Ghzzo, A. (000). Chage Sepaaton at a Plasma Edge n the Pesence of a Densty Gadent. Phys. Plasmas, Vol. 7, pp Shouc, M. (00). Numecal Smulaton of Plasma Edge Tubulence due to ExB Flow Velocty Shea. Czech. J. Phys., Vol. 5, pp.39-5

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23 00 Numecal Smulaton Fom Theoy to Industy Stozz, D., Shouc, M., Bes, A., Wllams, E.A. & Langdon, A.B. (006). Vlasov Smulatons of Tappng and Inhomogenety n aman Scatteng. J. Plasma Phys., Vol. 7, pp Waltz,.E., Candy, J., osenbluth, M.N. & Hnton, F.L. (999). Pogess on a Full adus Electomagnetc Gyoknetc Code. Poceedngs of EPS th Confeence on Cont. Fuson and Plasma Phys., ECA Vol. 3J, pp , Maastcht, Holland, June 4-8 Yanenko, N.N. (97). The Method of Factonal Steps, Spnge-Velag, New Yok.