GA A22703 POLOIDAL GRADIENTS, DIVERTOR DETACHMENT AND MARFES

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1 GA A22703 POLOIDAL GRADIENTS, DIVERTOR DETACHMENT by NOVEMBER 1997

2 DISCLAIMER This epot was pepaed as an account of wok sponsoed by an agency of the United States Govenment. Neithe the United States Govenment no any agency theeof, no any of thei employees, makes any waanty, expess o implied, o assumes any legal liability o esponsibility fo the accuacy, completeness, o usefulness of any infomation, appaatus, poduce, o pocess disclosed, o epesents that its use would not infinge pivately owned ights. Refeence heein to any specific commecial poduct, pocess, o sevice by tade name, tademak, manufactue, o othewise, does not necessaily constitute o imply its endosement, ecommendation, o favoing by the United States Govenment o any agency theeof. The views and opinions of authos expessed heein do not necessaily state o eflect those of the United States Govenment o any agency theeof.

3 GA A22703 POLOIDAL GRADIENTS, DIVERTOR DETACHMENT by This is a pepint of a pape to be pesented at the Sixth Intenational Wokshop on Plasma Edge Theoy in Fusion Devices, Septembe 15 17, 1997, Oxfod, United Kingdom and to be published in the Poceedings. Wok suppoted by the U.S. Depatment of Enegy unde Contact No. DE-AC03-89ER51114 GA PROJECT 3466 NOVEMBER 1997

4 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT Poloidal Pessue Gadients, Diveto Detachment and Mafes M.J. Schaffe Geneal Atomics, San Diego, Califonia , USA Abstact Because the adiation powe density fom a mafe scales appoximately as the squae of its plasma pessue, and since inceased adiation would aid diveto detachment fo high powe tokamaks, this pape identifies egions that might pemit locally inceased plasma pessue in steady state. The magnetic and dynamic (flow) constaints of magnetohydodynamics (MHD) ae examined fo self-consistent locally inceased pessue equilibia, in both the magnetically open tokamak scape-off laye (SOL) and the closed sufaces just inside the last closed flux suface. In most tokamak geometies it is difficult to ecycle paticles at a sufficient ate to sustain high pessue mafes, but they might be possible nea a diveto X point. 1. Intoduction Plasma in the magnetically open tokamak scape-off laye (SOL) and diveto obeys the equations of MHD equilibium with flow. Coss-field-static equilibium, in which only the flow paallel to B is apid, equies the pesence of Pfisch-Schlüte cuents in ode to satisfy J= 0. Expeimentally, SOL Pfisch-Schlüte cuents ae obseved to close though the electically conducting tagets of pesent tokamaks when the diveto plasmas ae attached [1]. The SOL poloidal pessue gadients, which ae localized in font of the tagets in attached opeation, ae equilibated by cuents cossing the tooidal magnetic magnetic field nomal to the magnetic sufaces. The Pfisch-Schlüte cuents flow paallel to B (foce-fee), both aound the common SOL and to the taget, to edistibute cuent fom the high B side to the low B side, whee exta cuent is needed to maintain equilibium in the weake magnetic field. When diveto plasmas detach in JET [1] and DIII D, the Pfisch-Schlüte taget cuents become vey small, possibly due to inceased taget plasma impedance. Despite the blocked diveto equilibium cuents, thee ae no expeimental signs of a loss of the coss-field-static equilibium, such as an enhanced poloidal flow fom the inne to the oute divetos in typical single-null top- o bottomdiveted tokamaks. Pevious wok showed that without taget cuents, coss-field-static equilibia in this geomety must have at least one poloidal pessue gadient egion on the SOL upsteam of the diveto gadients [2]. The pesent pape shows that poloidal pessue gadients ae allowed, but not equied, on closed magnetic sufaces as well. Howeve, because J= 0, cuent and pessue cannot be abitaily distibuted. Likewise, the GENERAL ATOMICS REPORT GA-A

5 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT poloidal pessue gadients, which dive paallel equilibium flows the dynamic pat of the MHD equilibium cannot be ignoed. Mafes commonly occu in o nea the X point, both inside and outside of the magnetic sepaatix duing detachment [3]. In this pape mafe means any localized egion of stongly adiating, low-tempeatue, high-density plasma, egadless of the mechanism(s) involved. Mafes ae usually assumed to exist with little o no pessue gadient on each affected magnetic suface. Howeve, some peliminay DIII D diveto Thomson scatteing data show locally inceased p e nea the X point, both in and outside the sepaatix. (Ion pessue cannot yet be measued thee.) It is woth consideing whethe mafes might be associated with any of the highe pessue equilibium egions, because, since a mafe adiates as n2 ~ p2 at a fixed tempeatue set by atomic pocesses, the inceased adiation would aid detachment at high tokamak powe. This pape discusses edge-plasma MHD constaints in detached plasmas and thei elations to the poloidal pessue distibution and to mafes. Only single-null divetos ae teated. 2. Magnetic Equilibia Simplified multispecies MHD equilibium equations with flow ae deived in [1] in a fom useful fo open and closed axisymmetic nested tooidal magnetic sufaces. In coss-fieldstatic equilibium only the paallel and poloidal momentum equations ae inteesting. In the tokamak limit the poloidal inetial tems ae negligible [1], and the poloidal equation simplifies to Bφ p I= JnBφ = p p. (1) Coodinate diections ae identified in Fig. 1, and p = e p. The plasma pessue is p, but subscipt p denotes a vecto component in the poloidal diection. The poloidal cuent steam function, I (,) z = I( xn, xp)= Bϕ µ 0, yields the divegence-fee cuent in the poloidal plane: Jn = 1 pi, Jp = 1 ni. Hee I is the poloidal cuent pe adian in the tous with I=0 at the majo axis and n= e n. Equation 1 states that a poloidal pessue gadient is equilibated by cuent cossing the tooidal magnetic field in the nomal diection. Futhe simplification follows if the poloidal pessue changes ae appoximated as steps and the J n as cuent sheets [2]. Then, BI φ p=, whee I = I= Jn dxp. (2) Also, let B R B φ = Figue 1 shows a detached SOL equilibium having inne and an oute egions with possibly diffeent constant pessues p in and p out, espectively. Pessue below the 2 GENERAL ATOMICS REPORT GA-A22703

6 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT z φ I mid R mid e n e p B φ p in p out I in R in I out R out Detached Fig.1. Detached single-null-diveted tokamak with elementay SOL cuent laye. detachment egion is small and is taken as zeo. Then, application of Eq. (2) to each pessue discontinuity yields RBI 0 0 in RBI RBI p p p p R 2 = in, out R 2 = out, mid R 2 = in out. (3) in out mid The nomally diected cuents satisfy J = 0 which globally is Iout + Imid Iin = 0. (4) Equations (3) and (4) can be solved to show pout pout + pin pin = ( ) ( ) 1 R2 out R2 2 in R2 1 mid R2 out + R2 2 in. (5) It is appaent fom Eq. (5) that p out p in except when R out = R in, i.e. when thee is no diveto. The pessue diffeence is lage if I mid is fee to flow at a adius R mid that makes the denominato small. If the left side of Eq. (5) is to be in its allowable ange (-1,1), then R mid is eithe inwad of R in o outwad of R out. The eason is that, when R mid < R in, I mid cosses a lage B φ and moe effectively equilibates pessue than I in, theeby supplementing the less effective I out [2]. The esulting equilibium has p in > p out. In the othe case, when R mid > R out, the sign of I mid eveses and p out > p in. Now I mid is less effective than even I out, but it combines with the moe effective I in to equilibate p out. These equilibia impose unequal inne and oute pessues duing detachment, especially when R out R in is lage, but they do not exihibit localized pessue nonunifomities unless the geomety is like Fig. 2. GENERAL ATOMICS REPORT GA-A

7 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT z φ (a) z φ I local R local p backgound (b) p backgound Bφ p local B φ I local R local p local I local R local p backgound Fig.2. Two examples with equal and opposite nomal cuents at the same adius. Figue 2 illustates two examples with the nomally diected cuents equal and opposite on a magnetic suface at diffeent poloidal locations at the same adius. In this case the cuents maintain a local pessue p local diffeent fom the pessue p backgound immediately adjacent on the same suface. The closed cuent loop stuctue automatically satisfies J = 0, and plocal pbackgound = R B Ilocal R2 0 0 local can take any value, positive o negative. Unlike the othe poloidal nonunifomities discussed in this pape, this one does not couple to any othe cuent. It couples to the equilibium only by the inetia of paallel flow. The figue 2 examples can couple, espectively, to mafes beside the X point and at the minimum majo adius. Figue 3 is analogous to Fig. 1, except it is fo closed magnetic sufaces. Application of Eq. (2) to each cuent sheet gives RBI RBI RBI p p p p p p R1 2 = , R2 2 = 12 23, R3 2 = (6) Equation (6) plus the J= 0 condition I1+ I2 + I3 = 0 yield ( ) + ( ) + ( ) =. (7) R1 2 R2 2 p12 R2 2 R3 2 p23 R3 2 R1 2 p31 0 Equation (7) constains less than Eq. (5), because thee is no efeence to zeo pessue. Equation (7) equies that the pessue bounded by the geatest and least of the thee cuent adii is intemediate between the othe two pessues. Fo a possible mafe at the p 31 location of Fig. (3), if R 2 < R 1 < R 3, then eithe p 31 > p 23 > p 12 o p 31 < p 23 < p 12, coesponding to high and low pessue mafes, espectively. If R 1 < R 3 < R 2, then eithe p 31 > p 12 > p 23 o p 31 < p 12 < p 23 again coesponding to high and low pessue mafes. But, if R 1 < R 2 < R 3, then p 31 is intemediate between the othe two pessues. 4 GENERAL ATOMICS REPORT GA-A22703

8 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT z φ I 2, R 2 p 23 e n e p B φ p 12 I 1, R 1 p 31 I 3, R 3 Fig.3. Elementay cuent laye with poloidal pessue gadients on a closed magnetic suface. 3. Paallel Equilibium and Plasma Flow We ae inteested in conditions unde which a mafe might exist at a significantly highe o lowe pessue than plasma elsewhee on the same magnetic suface. Even when the pesssue diffeence is equilibated poloidally by J n B φ, plasma still flows along B. The paallel component of the multispecies momentum equation is deived and discussed in [1]. Hee we simplify it to unifom B, equivalent to a staight magnetic flux tube, and two species: a single-fluid plasma and a fluid of chage-exchange (cx) neutals that on aveage exit the plasma with the plasma paallel velocity. Incoming low-velocity neutals do not affect the momentum balance. Then, if the cx neutal pessue is much smalle than the plasma pessue p, ( ) + dds( Γv p) Γ cx vw cx, (8) whee s = distance along B, Γ = ρv = plasma paallel mass flux, ρ and v ae its mass density and velocity, espectively, and Γ cx = ρ cx v cxn = nomal component of cx neutal flux exiting the plasma fom a laye of thickness w cx. Othe viscous tems, such as those due to poloidal flow, ae neglected. In the absence of nozzle expansion, the full set of equations limits M = v/c to < 1, as is well known. Now conside paallel flow out of a mafe. Let the mafe be fueled by ecycling neutals and poweed by a net deposition of enegy fom hotte plasma egions. The flow stats with Γ = 0 in the mafe. With no chage exchange, Eq. (8) gives the well known esult ( ) =γ (9) p1 p2 p2 M2 GENERAL ATOMICS REPORT GA-A

9 POLOIDAL PRESSURE GRADIENTS, DIVERTOR DETACHMENT whee p 1 and p 2 ae pessues inside and outside the mafe, espectively, γ is the specific heat atio, M = v/c = Mach numbe and c 2 = γ p/ρ. Fo M < 1 the mafe pessue is limited to about two times the pessue outside, egadless of fueling and heating, which affect the final Γ and v, espectivley. With chage exchange the pessue atio can be lage [4]. 4. Discussion and Conclusions The local high o low pessue zones of Fig. 2 match to typical mafe locations at the lowest adius egion of a magnetic suface (closed o open) and just inwad o outwad of the X point in the SOL [3]. Because J= 0 can be satisfied locally at these locations, thee ae no unexpected MHD equilibium constaints on mafe pessue. Whethe the mafe pessue can exceed the pessue on the est of the suface then depends just on the paticle flow. Application of Eq. (8) to DIII D (R 0 = 1.7 m, a = 0.6 m) yields a (not uneasonable) paticle souce equiement of Ṅ ~ 1021 s 1 to sustain p1 p2 ~ 2. Simila paticle souce consideations apply to the Fig. 3 closed-suface location that would be just above the X point. The main poblem is not achieving Ṅ, but how to etun so many paticles to the mafe. In most tokamak geometies the paticles must etun via lowe tempeatue and density SOL flows o even moe tenuous neutal flows, all of which ae flux limited. In fact, in a ecent numeical simulation of diveto detachment by the UEDGE code that had a gowing mafe nea the X point just inside the sepaatix [5], the mafe pessue was lowe. The paallel flow was into the mafe, and paticles exited acoss B. Howeve, mafes nea the X point (Fig. 2a) might be esupplied by the adjacent high ecycle paticle flux. To conclude, in most tokamak geometies it is difficult to ecycle paticles at a ate sufficient to sustain high pessue mafes, but they might exist nea a diveto X point. The detached configuation of Fig. 1 does not offe any unique localized pessue zones that might be matched to a mafe. Howeve, it specifies an in-out pessue diffeence that can be lage ( Rout 2 Rin 2 ~ 2 in low aspect tokamaks) and undoubtedly affects SOL and diveto equilibia quantitatively. This fundamental effect is not in any of the commonly used analytic and numeical models. Acknowledgments The autho had helpful discussions with MA Mahdavi, GD Pote and TW Petie. This wok was suppoted by U.S. Depatment of Enegy Contact DE-AC03-89ER Refeences [1] Schaffe MJ et al, Nucl. Fusion (1997) [2] Schaffe MJ, submitted to Comments on Plasma Phys. and Contolled Fusion (1997) [3] Petie TW et al, J. Nucl. Mateials (1992) [4] Knoll DA et al, Phys. Plasmas (1996) [5] Pote GD pivate communication (1997) 6 GENERAL ATOMICS REPORT GA-A22703