Approach to canonical pressure profiles in stellarators

Transcription

1 1 TH/P8-4 Aroach to canonical ressure rofiles in stellarators Yu.N. Dnestroskij 1), A.. Melniko 1), L.G. Elisee 1), S.E. Lysenko 1), K.A. Razumoa 1),.D. Pustoito 1), A. Fujisawa ), T. Minami ), and J.H. Harris 3) 1) Nuclear Fusion Institute, RRC "Kurchato Institute", Moscow, 1318, Russia, ) National Institute for Fusion Science, Toki, Jaan 3) Fusion Energy Diision, Oak Ridge National Laoratory, USA contact of main author: dnyn@nfi.kiae.ru Astract. Analysis of stellarator dataase hae shown that aleit on many deices the temerature rofile consistency is asent, howeer, on seeral deices the ressure rofiles turn out to e selfconsistent. The ressure rofile in the L-mode may e fitted y quasi-linear function: -1 d/d ~ const = k = 1.3 +/-.1. To descrie the ressure self-consistency we use the ariation rocedure, reiously used for tokamaks. We consider the ariation rolem to minimize the energy functional W under the constraint that the total lasma current J is a constant. As a result, we otained the equations for canonical equilirium and corresonding ressure rofiles in L- and H-modes with low, moderate and high magnetic shear. Maximal sloes of these rofiles are close to maximal sloes of exerimental ressure rofiles osered in TJ-II. 1. Introduction It is well known that the temerature rofiles in a tokamak are self-consistent [1-3]. For the stellarators there is no temerature rofile consistency [4]. The ressure rofile consistency concet works in tokamaks [5]. There was found one examle of the ressure rofile stiffness in the ellet injection ersus gas uff exeriment in LHD [6], and also the features of the ressure rofile consistency in other LHD exeriments [7]. arious examles of ressure rofile self-consistency for seeral stellarators were resented in [8]. In this reort we extend exerimental dataase on ressure self-consistency and use the ariational rocedure deeloed in [9] for the analysis of the existence of the referred (canonical) ressure rofiles in stellarators. Plasma temerature and density rofile eolution hae een considered for the arious exeriment. NBI heating of on- and off-axis ECRH heated lasma on TJ-II [1], ECRH ower scan on W7-AS [11] and CHS [1], high T i mode on CHS [13], on- and off-axis ECRH on W7-AS [4] and gas uffing on ATF [14] were osered (Tale 1). In the TJ-II stellarator, NBI heating (P NBI = 3 kw) of the target ECRH lasma (P ECRH = 3 kw) leads to dramatic changes of the lasma density and temerature. n e and T e rofile eolution measured y high resolution Thomson Scattering diagnostic is shown in Fig. 1. The alues aried u to an order of magnitude, (.3 < n e () < m -3,. < T e () < 1 ke), the rofiles aried from hollow to eaked (density), and from eaked to flat (electron temerature). In site of the large difference in n e and T e rofiles in the analyzed regime, their roduct, the lasma ressure e, resents much stronger rofile resilience in the confinement zone of the lasma column. It was found that the normalized ressure rofiles norm = ()/() are much less scattered in comarison with lasma n e and T e rofiles, see Fig.. In the CHS exeriments with on-axis P EC = kw and density n e = m -3 ariation [1], the similar ehaior was found: the increase of T e was accomanied y the decrease of n e, leaing norm () ractically unchanged. In the exeriments with the standard major lasma axis (R ax =9.1cm, n e ()= m -3, T e ()~T i ()=5 e), and otimized one

2 TH/P8-4 (R ax =87.7cm, n e ()= 1 19 m -3, T e ()= e, T i ()=13 e) [13], the same tendency was found, P norm remains almost constant. In high T i mode (n e ()= m -3, T e ()=7 e, T i ()=1 ke) [13], norm almost coincides with the rest of discussed CHS rofiles. In W7-AS exeriment with on-axis P EC ariation from. to.8 MW at almost the same density n e ~ 1 19 m -3 [11], the similar ehaior was found: increase of T e was accomanied y a decrease of n e, remaining norm () ractically unchanged. In another exeriment, on- and off-axis ECRH alternate with corresonding T e and n e ariations [4]. It is highlighted in [4] that during off-axis ECRH (=.6) the central density is eaking without an additional article source, which leaded to almost unchanged norm (). At the gas uffing in the ATF [14], the lasma density rise was accomanied y the concordant decay of T e, again remaining norm () ractically unchanged. Desite the difference in the magnetic configurations (heliac TJ-II, torsatrons CHS and ATF, otimized W7-AS), a remarkale similarity is seen in the normalized ressure rofiles; in other words, the normalized ressure rofile has uniersal shae for normal confinement (Lmode) in all the osered exeriments (Fig. 3). The uniersal rofiles can e fitted y a quasilinear function in the confinement zone (.<<.8), -1 d/d ~const=k = -1 linear / linear, where linear is the radial extension, where rofile has almost linear shae In the osered cases linear ~.6, k 1.3 ±.1 (see TABLE 1). Contrary to the L-mode, ressure rofiles show different shaes during imroed confinement modes. An examle of the edge transort arrier is shown in Fig (); norm () in HDH confinement mode in W7-AS strongly differs from the uniersal rofile, while the reference Normal Confinement rofile elongs to the uniersal one [15]. In case of the ITB formation the ressure rofiles hae clearly two comonents. Outside the ITB, the ressure rofiles show strong similarity (the uniersal rofile takes lace), while in the ITB area -1 d/d is significantly higher. Figure 3 shows W7-AS data with on-axis P EC =1. MW, where the temerature and ressure rofiles show the ITB formation at ~.5 [11]. In the CHS the more ronounced ITB was otained for the high ower on-axis ECRH at ~.4 [13]. Again, outside the ITB area the rofiles coincide with the uniersal one. TABLE 1. FITTING RESULTS OBTAINED FROM DIFFERENT STELLARATORS. k is linear regression error for k. Deice Tye R, m a, m B, T ι/π(a) -k k Regime, [Ref.] TJ-II heliac , low shear EC on axis+nbi [1] EC off- +NBI [1] W7- AS modular coils low shear, P EC scan, n e const [11] EC on- off-axis [4].13 CHS torsatron high shear HDH [15] EC on- +NBI [1] EC on-, ITB [13] High T i [13] ATF torsatron shear gas uff [14]

3 3 TH/P On-axis ECRH 1. Off-axis ECRH.8 T e (ke).6.4 T e.5. time n e (1 19 m -3 ) 3 1 On-axis ECRH time n e time Off-axis ECRH time FIG. 1. T e (uer) and n e (lower) rofile eolution in the NBI exeriments for on-axis (left) and offaxis (right) ECRH in TJ-II. norm e a) EC off-axis EC on-axis norm FIG. a) TJ-II normalized ressure rofiles for data from FIG.1, ) norm () for the data resented in TABLE forms the uniersal rofile; CHS: red dots high Ti mode, rown R ax ariation, red P EC ariation, ATF: urle - gas uff, W7-AS: green - P EC -on scan, lue EC on- and off-axis, lack HDH-mode, dark lue reference Normal Confinement mode, TJ-II: lue dots EC on, off. Profiles from all machines and exeriments were radially normalized to the actual lasma size, norm (1)=. ) norm FIG. 3. The uniersal rofile exists outside the ITB area. CHS: fat lue high Ti mode, fat red ITB, W7-AS: green PEC =.-.8 MW, thin lue ITB (PEC=1. MW). Dashed lines designates internal transort arrier.

4 4 TH/P8-4 To summarize the emirical oseration, we may conclude: (I) In site of large ariety in the rofiles of lasma electron temerature and density, their roduct, the electron ressure resents the feature of rofile constancy in stellarator deises in osered exeriments with wide range of the lasma and heating arameters. (II) In the L-mode lasmas of the medium size stellarators the ressure rofiles show remarkale similarities etween each other in low (TJ-II, W7-AS) and high (CHS, ATF) magnetic shear configurations. So, the uniersal rofile, characterized y -1 d/d ~ const = k, was found for L-mode lasmas and out of the ITB area. The other tyes of rofile like LHD case [6] may take lace for secific lasma conditions. (III) The oseration of the uniersal constant k in the L-mode (e.g. in the asence of strong E B effects) may suggests that the turulence and the associated transort reach some kind of saturation leel, which does not deend on the asolute alues of T e and n e, ut alternatie exlanations (e.g. ased on the role of atomic hysics and links etween magnetic configuration and gradient) cannot e excluded The alidation of the transort-ased hyothesis would require to characterize the link etween local gradients and turulent transort.. Canonical ressure rofiles We start from the energy integral B W = + d, (1) γ 1 where B is the magnetic field, is the lasma ressure, γ is the ratio of secific heats, and the integration is erformed oer the lasma olume. First, we integrate here oer the toroidal angle ζ, so that W will e transformed into the integral oer the axially symmetric toroid inside the toroidally aeraged lasma oundary. In conentional stellarators with lanar circular axis, all hysical quantities can e reresented as f = f + f () ~ where f ( r, z) is the axially symmetric (or toroidally aeraged) art of f, and f ( r, z, ζ ) is its oscillating art, ( r, z, ζ ) are the usual cylindrical coordinates with z-axis directed along the main axis of the system. Accordingly, B= B+ B, B = Bteζ + B, (3) where B is the axially symmetric and B is the helical magnetic field, e is the unit ector in the toroidal direction, B t is the toroidal comonent of B and B is the oloidal magnetic field. It is shown in [17] that ( B ) d = ~ B B d. (4) With this relation we otain W = B B + γ 1 d, (5) where is the axially symmetric art of, or aeraged lasma ressure. To simlify the notations we further use instead of. In (5) the integration is erformed oer the D quasi-tokamak region. ζ

5 5 TH/P8-4 To calculate the ariation of the energy, δw, we need δ B. The axially symmetric oloidal field can e descried as B = rotate ζ. Therefore B δ = diδat ( eζ B ) + jζ δat, (6) where jζ is the ζ -aeraged toroidal comonent of the current density: j ζ = e ζ rotb. With (6) we otain from (5) δ δw = BtδBt + jζ δat + d. (7) γ 1 We assumed here that the helical field is fixed (is not aried). In stellarators we hae [17] 1 ψ ψ B = [ ( ψ ψ ) ζ ], ra t =. (8) π π Here ψ ν, is the oloidal flux of the helical field B ~. In [16] the energy was minimized under the constraint that the total lasma current is a constant of motion, δj=. Following this aroach we consider the ariation rolem for the functional Φ = W λj, (9) where λ is the Lagrange multilier. The net toroidal current is jζ J = j ds = jζ ds = d, (1) πr S S where S is the toroidal cross-section of the lasma and S is its ζ-aeraged image, with d = πrds. Then jζ δjζ δ δw λδj = BtδBt + δ ( rat ) λ + d, (11) r πr γ 1 and the ariation rincile with (9) gies the following Euler equation: B jζ λ j t ζ ( ψ ) Bt + + =. (1) ψ πr πr ψ γ 1 where rime means the deriatie with resect to ψ. ~ In stellarators with B / << 1 we hae [17] B t Here F( ψ ) π r B t j ζ, (13) πr dψ FF ( ψ ) d = and = + πr ( 1+ Ω ) ~ Ω = B / B with ζ ζ... staying for the toroidal aeraging, and B is the toroidal magnetic field at the axis. The quantity Ω can lay an imortant role in some secial cases when it can e as large as the inerse asect ratio with strong in-out asymmetry on the magnetic surfaces. But usually Ω <<1 so it can e disregarded in (13). With (13) equation (1) turns into FF λ( FF ) + ( λ )(1 + Ω ) + =, (14) 4π r γ 1 The second and the third terms in (14) are constant at the magnetic surfaces while the first term is not constant if nominator is not equal to zero. So disregarding Ω we otain finally FF λ ( FF ) =, γ λ( γ 1) =, (15) We ut further that γ = 5/3, γ/(γ-1) = 5/. The equations (15,1)-(16,) imly that

6 6 TH/P8-4 FF = С F ex(ψ/λ), = C ex(.5ψ / λ). (16) With these functions the two-dimensional equilirium equation for stellarators [17] ecomes ( ψ ψ ) ex(ψ / λ) di = πj / r = 4π C ex(.5 / ) C ψ λ F. (17) r r The latter equation can e naturally called Canonical equilirium equation for conentional stellarators. The equation (17) is of Poisson tye equation, which requires the oundary condition at the lasma oundary S, which is a magnetic surface, ψ(s) = const = ψ. (18) Equation (17) includes three arameters, С F, С and λ, so we hae to add three additional conditions to find them. One can e the rescrition of ψ at the magnetic axis, ψ(m ) = ψ, (19) or this can e relaced y the gien alue of the rotational transform =-dψ/dφ at the magnetic axis. Here Φ = B ds t is the toroidal flux inside the magnetic surface ψ. S ψ The second constraint is the gien total lasma current: J = j d S. () S In stellarators this is usually small, and often J = is assumed. In exeriments we usually know the aeraged β (the ratio of lasma ressure to magnetic ressure). It can also e reresented in the model y β = / B, (1) where is the lasma ressure at the magnetic axis and B is the toroidal field at the axis. Prescrition of β can e the third required condition. Comlete solution of the canonical equation (17) would gie us the arameter λ, which was introduced as the Lagrange multilier in (9). With known λ we could find the rofiles of F and, which for the canonical equilirium are determined y the one-dimensional equations (16). The solution for ressure can e written as 1.5ku 1.5k e e = ( ) + 1.5k, () 1 e where and are the lasma ressures at the magnetic axis and at the edge, resectiely, ψ ψ u = ( < u < 1) = ( ψ ψ ) ψ ψ k. (3) λ A general solution to Eq. (17) is ψ = ψ + ψ + ψ, (4) ext l where ψ ext descries the contriution due to the external oloidal field and ψ l due to the equilirium currents in the lasma. We assume here ψ = and consider the case ψ l << ψ, which corresonds to low-β lasma. Then ψ ψ. (5) Let us consider the simlest case of the cylindrical lasma with circular cross-section. In this case ψ ψ (), where is the radial coordinate of the lasma cross-section. Then = ψ / = πb, (6) where is the rotational transform which, in a general case, can e aroximated y ext

7 7 TH/P8-4 ( ξ = + ) with = (), = (1), ξ = / and is the minor radius, so that ξ 1. This yields ψ [ + ( ) / ] = πb ξ ξ u = ξ ( ) + + (7) ξ. (8) This can e called the low-β aroximation of u. With gien / and u we hae the last unknown k. As a articular examle let us consider 1.5k the case with the rescried ratio / = e i.e k = - 4/5 ln( / ). This choice corresonds to the following oundary condition 1 d ( u = 1) = 1.5k, (9) du Such a tye of oundary condition was roosed for tokamaks y Kadomtse in his interretation of the ariation rocedure [18]. The condition (9) gies = ex(1.5ku). (3) Exression (3) with u gien y (8) reduces to c( ξ ) = ex{ [ln( ) /(1 + )] ξ [1 + ( 1) ξ / ]} (31) which can e called as a Kadomtse-tye rofile for stellarators. We consider the models of H- and L-modes, using the freedom in selecting the arameter / descriing the lasma, / = 1 (for the H-mode), / = 1 (for the L-mode), (3) and 3 cases with different shear, which is a characteristic of a stellarator configuration / = 1 (zero shear), / = (moderate shear), / = 4 (large shear) (33) Sustituting (3)-(33) into (31) we can find the largest sloe coefficients k = max [-(1/ )d c /dξ] of the rofiles of (ξ)/. The results are shown in figure 3. This shows 1.45 < k < 1.55 for the L-mode and k = 1.3 for the H-mode. It was shown earlier in Tale 1 that k = 1.3 ±.1 for the exerimentally measured L-mode ressure rofiles in seeral deices. The roximity of these exerimental results and the aoe theoretical estimations lead to the conclusion that the oserale exerimental self-consistency of ressure rofiles in stellarators might e ased on the minimum energy ariation rincile. Note that the reresentation of ressure rofiles in the exonential form similar to (31) was used recently during the analysis of the LHD exerimental rofiles [1]. 1.5 L-mode, zero shear: = =1 moderate shear: =1, = large shear: =1, m =4 k=1.55 k=1.46 k=1.5 / 1..5 L-mode zero shear moderate shear H-mode, moderate shear k=1.3 large shear ξ=/ FIG. 4. The canonical ressure rofiles for stellarators with different magnetic configuration. Here k= max [-(1/ )d c /dξ].

8 8 TH/P Conclusion It was assumed for a long time that the energy and article transort in a stellarator is defined in main y neoclassical transort coefficients and in this sense the stellarator differs radically from a tokamak, where the transort is anomalous and defined y the lasma turulence. Neertheless, gradually, the exerimental facts originated the dout in such a aradigm. The oseration of the H-mode and internal transort arriers and the roximity of the stellarator energy confinement time scaling to the tokamak scaling led to the oinion that the turulent transort lays the imortant role in stellarators. It was reorted recently [7] that the ressure rofile self-consistency is osered in stellarators, as well. In this Reort the ariation rocedure (roosed years ago for tokamaks) is used to construct the ressure canonical rofiles for stellarators. As a result we come to the twodimensional equilirium equation for so called canonical equilirium. The corresonding ressure rofiles are estimated in low-β lasma aroximation. The rofiles thus otained are close to the exerimental normalized ressure rofiles, although slightly differ from them as in tokamak case. Howeer, in tokamaks the temerature rofiles of electrons and ions are self-consistent also, which does not seen so far in stellarators. Aarently, this difference is exlained y the asence of the total current in stellarator and, as a consequence, y a smaller influence of Ohm s law. As a result in a stellarator the temerature rofile easily changes its form ut this feature is quite limited in a tokamak. The existing of two self-consistent rofiles in a tokamak (temerature and ressure) allows one to construct the transort model concerning the temeratures and lasma density ased on this roerty [5,19]. Unfortunately the second self-consistent rofile for the stellarator is not seen so far, so the rolem of full canonical rofiles transort model for stellarator remains oen. Acknowledgments Kurchato team was suorted y Grants RFBR and 7--11, INTAS Authors thank C. Hidalgo and F. Wagner for useful discussions. References [1] COPPI, B., Comments Plasma Phys. Control. Fusion 5 (198) 61. [] ESIPTCHUK, Yu.., RAZUMOA, K.A., Plasma Phys. Control. Fusion 8 (1986) 153. [3] RYTER, F., et al., Plasma Phys. Control. Fusion 43 (1) A33. [4] WAGNER, F., et al., Phys. Plasmas 1 (5) 759. [5] DNESTROSKIJ, Yu.N. et al., Nucl. Fusion 46 (6)953 [6] YAMADA, H. et al., Nucl. Fusion 41 (1) 91. [7] MIYAZAWA, J. J. Plasma Fusion Res. 81 (5) 3. [8] MELNIKO, A et al., 7 34-th EPS Plasma Physics Conf. Re. P.6. [9] DNESTROSKIJ, Yu.N. et al., sumitted to Plasma Phys. Control. Fusion. [1] LINIERS, M. et al., 15 ISW, Madrid, 5, P3-14. [11] ERCKMANN,. et al., In: Radio Frequency Power in Plasmas, AIP (1999) [1] NAKANO, H. Priate communication. [13] OKAMURA, S. et al., Nucl. Fusion 39 (1999) 1337 [14] SHATS, M. et al., Phys. Plasmas (1995) 398. [15] MCCORMICK, K., Phys. Re. Lett. 89 () 151 [16] HSU, J.Y., CHU, M.S., Phys. Fluids 3 (1987) 11. [17] PUSTOITO,.D., in Reiews of Plasma Physics (KADOMTSE, B.B. and SHAFRANO,.D. Ed.), Consultants Bureau, New York) 1 1. [18] KADOMTSE, B.B., So. J. Plasma Phys. 13 (1987) 443. [19] DNESTROSKIJ, Yu. N. et al., Plasma Phys. Control. Fusion 49 (7) 1477.