Tokamak elongation how much is too much? I Theory

Transcription

1 Tokamak elongation how much is too much? I Theory J. P. Freidberg 1, A. Cerfon 2, J. P. Lee 1,2 Abstract In this and the accomanying aer the roblem of the maximally achievable elongation κ in a tokamak is investigated. The work reresents an extension of many earlier studies, which were often focused on determining κ limits due to (1) natural elongation in a simle alied ure vertical field or (2) axisymmetric stability in the resence of a erfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum ractical elongation limit in a real exeriment. A basic resistive wall stability arameter γτ w is introduced to model the feedback system which although simle in aearance actually catures the essence of the feedback system. Elongation limits in the resence of feedback are then determined by calculating the maximum κ against n = resistive wall modes for fixed γτ w. The results are obtained by means of a general formulation culminating in a variational rincile which is articularly amenable to numerical analysis. The rincile is valid for arbitrary rofiles but simlifies significantly for the Solov'ev rofiles, effectively reducing the 2-D stability roblem into a 1-D roblem. The accomanying aer rovides the numerical results and leads to a shar answer of how much elongation is too much? 1. Plasma Science and Fusion Center, MIT, Cambridge MA 2. Courant Institute of Mathematical Sciences, NYU, New York City NY 1

2 1. Introduction It has been known for many years that tokamak erformance, as measured by ressure and energy confinement time, imroves substantially as the lasma cross section becomes more elongated. There are, however, also well known limits on the maximum achievable elongation, which arise from the excitation of n = vertical instabilities. When designing next generation reactor scale tokamak exeriments [Aymar et al. 22; Najmabadi et al. 1997; Najmabadi et al. 26; Sorbom et al. 215], where high erformance is critical, it is thus imortant to be able to accurately redict the maximum achievable elongation κ as a function of inverse asect ratio ε = a / R, where a is the minor radius of the device, and R is the major radius. The inverse asect ratio is a articularly imortant arameter since different reactor designs have substantially different values for this quantity. Secifically, standard and high field tokamak reactor designs have 1 / ε [Aymar et al. 22; Najmabadi et al. 1997; Najmabadi et al. 26; Sorbom et al. 215; Schissel et al. 1991; Hutchinson et al. 1994], while sherical tokamaks have smaller 1 / ε 1.5 [Sabbagh et al. 21; Sykes et al. 21]. Since otimized lasma erformance and corresonding minimized cost deend strongly on κ it is imortant to have an accurate determination of κ = κ(ε) and this is the goal of the resent and accomanying aer. The roblem of determining κ = κ(ε) has received considerable attention in ast studies but, as discussed below, there is still an imortant ga in our knowledge. To ut the roblem in ersective we note that earlier studies generally fall into one of three main categories, each one roviding valuable information, but not the whole story. These are summarized as follows. The first class of studies involves the concet of natural elongation. Here, the tokamak is immersed in a deliberately simle external oloidal magnetic field, usually a ure vertical field. The lasma is exected to be stable against n = vertical instabilities since one vertical osition is the same as any other. In these studies [Peng and Strickler 1986; Roberto and Galvão 1992] the free boundary equilibrium surface is calculated (i.e. the elongation κ and the triangularity δ are determined) for a range of ε from which it is straightforward to extract the desired κ = κ(ε). Excellent hysical insight is rovided by these calculations. Even so, the maximum redicted natural elongations are substantially smaller than those achieved in exeriments. The reason is that exeriments have feedback systems that can stabilize n = vertical instabilities, thereby allowing larger κ s than those redicted by marginally stable naturally elongated configurations. The second class of studies involves the calculation of the critical normalized wall radius b/ a for an ideal erfectly conducting conformal wall required to stabilize a desired elongation; that is the analysis basically determines b /a = f (ε,κ). Here too, the studies [Wesson 1978] (and references therein) rovide valuable insight. Secifically, the 2

3 studies determine the maximum b/ a that might otentially be able to be stabilized by feedback. However, the analysis does not actually redict whether or not a ractical feedback system can be constructed to rovide stability. Equally imortant, from a ractical oint of view the actual value of b/ a does not vary much from exeriment to exeriment. The third class of studies consists of detailed, engineering level designs that redict the maximum elongation and which include many effects such as lasma rofiles, real geometry, safety margins, and most imortantly, engineering roerties of the feedback system. Such studies [Kessel et al. 26] are very realistic and are exactly what is required to design an actual exeriment. Nevertheless, these are oint designs whose main goals are focused on a secific machine rather than roviding scaling insight in the form of κ = κ(ε). The resent analysis attemts to fill an imortant ga in our knowledge, namely determining the maximum elongation as a function of inverse asect ratio including the constraints arising from the feedback system. The calculation is largely analytic combined with some straightforward numerics. In our model the lasma is assumed to be u-down symmetric and is characterized by the following arameters: inverse asect ratio ε = a / R, elongation κ, triangularity δ, oloidal beta β, normalized wall radius b / a, and an aroriately defined arameter reresenting the caabilities of the feedback system. Defining this feedback arameter and including it in the analysis is the main new contribution of the research. A simle but reliable definition of the feedback arameter is based on the following two observations. First, ractical feedback is feasible when the growth rate γ of the n = vertical instability is small. Since the lasma is surrounded by a finite conductivity wall the instability of interest is actually a resistive wall mode. The second observation is that if the feedback coils are located outside the resistive wall, as they usually are, then an effective feedback system must have a raid field diffusion time τ w. This is imortant because once an unstable lasma motion is detected the feedback resonse fields must quickly diffuse through the wall in order to reach the lasma. In other words τ w should be short. In rincile it is ossible to trade off growth rate in favor of resonse time or viceversa. However, the overall effectiveness of the feedback system is deendent uon the combined smallness of γ and τ w. The conclusion is that a simle arameter that takes into account the feedback system is the roduct of these two quantities: γτ w = feedback caability arameter (1) A good way to think about this arameter is as follows. In designing a new exeriment the roerties of the feedback system deend mainly on the geometry of the 3

4 vacuum chamber, the availability of fast-resonse ower sulies, the maximum feedback ower available, the number of feedback coils, and the sensitivity of the detectors. These reresent a combination of engineering and economic constraints that should not vary much as ε changes. Consequently, the engineering value of γτ w is a good measure of the feedback caability. It reresents the maximum value of the resistive wall growth rate that can be feedback stabilized. Now, consider the design of a new exeriment with ε as a arameter over which otimization is to be erformed. For a fixed ε the lasma elongation should be increased until the growth rate of the n = resistive wall mode is equal to the value of the feedback caability arameter given in Eq. (1). In this rocess, the triangularity δ may also be adjusted to find the maximum allowable κ. This resulting κ is the elongation that maximizes erformance for the given ε. By reeating the rocedure for different ε it is then ossible to determine the otimum asect ratio, including the effects of transort, heating, magnetic design, etc., that results in the eak value of maximum erformance and corresonding minimizes cost. In our analysis ractical engineering values of γτ w are chosen by examining the data from existing exeriments as well as ITER [Aymar et al. 22]. Based on this discussion we can state that the goals of the analysis are to determine the maximum allowable elongation κ and corresonding triangularity δ as a function of inverse asect ratio ε subject to the constraints of fixed oloidal beta β, normalized wall radius b / a, and feedback arameter γτ w ; that is we want to determine κ = κ(ε;β,b /a, γτ w ) δ = δ(ε;β,b /a, γτ w ) (2) This aer resents the analytic theory necessary to determine these quantities. The analysis is valid for arbitrary lasma rofiles. An additional useful result resented as the derivation rogresses is an exlicit relationshi between vertical stability and neighboring equilibria of the Grad-Shafranov equation, a relationshi long believed to be true but to the authors knowledge, never exlicitly aearing in the literature. The second aer in the two art series resents the numerical results. For numerical simlicity the results are obtained for the Solov'ev rofiles [Solov ev 1968] although with some additional equilibrium numerical work it is ossible to include arbitrary rofiles. One may exect Solov ev rofiles to rovide reliable hysical insight since axisymmetric MHD modes are thought to not deend too sensitively on the details of the current rofile [Bernard et al. 1978]. The overall conclusions are that the maximum elongation (1) decreases substantially as ε becomes smaller and (2) is substantially higher than that redicted by natural elongation calculations, much closer to what is observed exerimentally. The maximum elongation is weakly deendent on 4

5 β and also does not vary much with b /a rimarily because this quantity itself has only limited variation in ractical designs. There is, however, a substantial increase in maximum elongation as γτ w increases. As exected, feedback is very effective in increasing elongation and overall lasma erformance. A slightly more subtle effect is the value of the otimum triangularity corresonding to maximum elongation, which is noticeably smaller than that observed exerimentally in high erformance lasma discharges. The reason is resumably associated with the fact that maximum overall erformance deends on turbulent energy transort as well as MHD stability. Although turbulent transort is known to be reduced with increasing triangularity, which hels exlain the data, it is unfortunate that the current emirical scaling laws do not exlicitly include this effect. The analysis is now ready to roceed. 2. Equilibrium The equilibrium of an axisymmetric tokamak is described by the well-known Grad- Shafranov equation [Grad and Rubin 1958; Shafranov 1958] d Δ * Ψ = µ R 2 dψ 1 df 2 2 dψ (3) Here, Ψ(R,Z) = oloidal flux/2π and (Ψ), F(Ψ) are two free functions. The Δ * oerator, lus the magnetic field B = B + B φ e φ and current density J = J +J φ e φ are given by Δ * Ψ R 1 Ψ R R R + 2 Ψ Z 2 B = 1 R Ψ e φ + F R e φ (4) µ J = df dψ B 1 R Δ* Ψ e φ For the study of n = resistive wall modes the essential hysics is catured by considering u-down symmetric equilibria and this is the strategy adoted here. Also, great analytic simlicity follows by choosing the free functions to corresond to the Solov'ev rofiles [Solov ev 1968] for which F F (Ψ) =C 1 and (Ψ) =C 2, with rime denoting d /dψ. Only two constants, C 1 and C 2, are needed to secify an equilibrium. 5

6 The Grad-Shafranov equation can now be conveniently normalized in terms of two equivalent constants, Ψ and A, one of which ( Ψ ) scales out entirely from the final formulation. The full set of normalizations is given by R = R X Z = R Y Ψ = Ψ Ψ (5) µ d dψ = Ψ (1 A) R 4 F df dψ = Ψ A R 2 Note that the normalized flux has an uer case italic font. In terms of these normalizations the Grad-Shafranov equation lus the critical field quantities needed for the analysis reduce to Ψ XX 1 X Ψ X + Ψ YY = A +(1 A)X 2 (R 2 / Ψ )B = 1 ( X Ψ e + Ψ e Y R X Z ) (6) (R 3 / Ψ )µ J φ = 1 X A +(1 A)X 2 The boundary conditions require regularity in the lasma and Ψ(S P ) = with S P the lasma boundary. This imlies that Ψ(X,Y ) < in the lasma volume. Tyical values of the free constant A in decreasing order are as follows A = 1 A = (1 ε)2 A = 1 (1 ε) 2 (Ψ) = (force free) F(Ψ) = R B (vacuum toroidal field) J φ (1 ε,) = (inner edge current reversal) (7) The task now is to find a solution for Ψ(X,Y ). Exact solutions to the Grad- Shafranov equation for Solov'ev rofiles have been derived by a number of authors [Zheng et al. 1996; Weening 2; Shi 25]. Here, we follow the formulation of 6

7 reference [Cerfon and Freidberg 21; Freidberg 214]. Secifically, an exact solution to Eq. (6) can be written as Ψ ( X,Y ) = X 4 Ψ 1 = 1 Ψ 2 = X A 1 2 X 2 ln X X 4 8 +c Ψ +c Ψ +cψ +c Ψ +c Ψ +c Ψ +c Ψ Ψ 3 =Y 2 X 2 ln X (8) Ψ 4 = X 4 4X 2 Y 2 Ψ 5 = 2Y 4 9X 2 Y 2 + 3X 4 ln X 12X 2 Y 2 ln X Ψ 6 = X 6 12X 4 Y 2 + 8X 2 Y 4 Ψ 7 = 8Y 6 14X 2 Y X 4 Y 2 15X 6 ln X +18X 4 Y 2 ln X 12X 2 Y 4 ln X The free constants c j are chosen to match as closely as ossible the well-known analytic model for a smooth, elongated, D shaed boundary cross section arametrically in terms of τ by [Miller et al. 1998] S given P X = 1 + ε cos(τ + δ sin τ) Y = εκ sin τ (9) Here, ε = a / R is the inverse asect ratio, κ is the elongation, and δ = sin δ is the triangularity. The geometry is illustrated in Fig. 1. In ractice the j c are determined by requiring that the exact lasma surface Ψ =, lus its sloe and curvature match the model surface at three oints: (1) the outer equatorial oint, (2) the inner equatorial oint, and (3) the high oint maximum. Taking into account the u-down symmetry we see that the requirements translate to 7

8 Ψ(1 + ε,) = outer equatorial oint Ψ(1 ε,) = Ψ(1 δε,κε) = Ψ X (1 δε,κε) = Ψ YY (1 + ε,) = N 1 Ψ X (1 + ε,) Ψ YY (1 ε,) = N 2 Ψ X (1 ε,) Ψ XX (1 δε,κε) = N 3 Ψ Y (1 δε,κε) inner equatorial oint high oint high oint maximum outer equatorial oint curvature inner equatorial oint curvature high oint curvature (1) where for the given model surface, ( ) 2 N 1 = d 2 X dy 2 = 1 + δ τ= εκ 2 ( ) 2 N 2 = d 2 X dy 2 = 1 δ τ=π εκ 2 N 3 = d 2 Y dx 2 = κ τ=π/2 ε cos 2 δ (11) The determination of the c j has been reduced to finding the solution to a set of seven linear, inhomogeneous algebraic equations, a very simle numerical roblem. Hereafter, we assume that the c j have been determined, thereby comletely defining the equilibrium solution. A tyical set of flux surfaces corresonding to the high β, tight asect ratio NSTX sherical tokamak is illustrated in Fig. 2, where A has been chosen so that J φ (1 ε,) =. The surfaces look quite reasonable and as exected, exhibit a large shift of the magnetic axis. The one final iece of information required for the resistive wall stability analysis is the value of oloidal beta β. In general β and the toroidal current I can be exressed in terms of the free constants A, Ψ. The advantage of the normalizations introduced by Eq. (5) is that β is only a function of A but not Ψ. This can be seen by noting that the average ressure and toroidal current can be written as, Average ressure: 8

9 dr dr 3 V = dr = 2πR = 2πΨ 2 (1 A) µ R V X dx dy ΨX dx dy (12) Toroidal current: µ I = B dl! = µ J φ Ψ= ds = Ψ 1 R X A +(1 A)X 2 dxdy (13) Ψ Ψ where B is the magnitude of the oloidal magnetic field. We now define β as β 2µ B 2 B = L P =! Ψ=! Ψ=! Ψ= B dl dl dl = µ I L P (14) Substituting for and I yields β = 4πR L 2 P V Ψ (1 A) ΨX dx dy Ψ 1 (15) 2 X A +(1 A)X 2 dx dy Since Ψ is a function of A but not Ψ it follows that as stated β = β (A,ε,κ,δ). Note also that the volume V and oloidal lasma circumference L are urely geometric P factors that can be calculated from the solution given by Eq. (8) once the c j have been determined. Furthermore, with our normalizations the ratio R L P 2 R and Ψ ; that is R L 2 P equilibrium. /V is indeendent of /V = f (A,ε,κ,δ). This comletes the discussion of the 9

10 3. Resistive wall stability formulation In this section we derive a variational rincile describing the stability of n = resistive wall modes in a tokamak. The analysis starts from the ideal MHD normal mode equation for lasma stability. To obtain the variational rincile we will need to decomose the volume surrounding the lasma into three regions: an interior vacuum region, a thin wall, and an exterior vacuum region [Haney and Freidberg 1989]. The final variational rincile is exressed in terms of three erturbed flux functions for the (1) lasma, (2) interior vacuum region and (3) exterior vacuum region. To obtain this variational rincile use is made of two natural boundary conditions. The final ste in the formulation is to derive relationshis between the two erturbed vacuum fluxes and the erturbed lasma flux, a task accomlished by the alication of Green s theorem. The analysis is somewhat simlified by focusing on udown symmetric equilibria and considering only vertical-tye dislacements which are the most dangerous exerimentally. The final form of the variational rincile is straightforward to evaluate numerically and most imortantly directly takes into account the feedback constraint γτ w = constant as described in the Introduction. The variational formulation, when imlemented numerically, allows us to determine the maximum elongation and corresonding triangularity as a function of asect ratio. The basic ideal MHD stability equations The starting oint for the analysis is the ideal MHD linear stability equations for the lasma given by ω 2 ρξ + F(ξ) = F(ξ) = J 1 B + J B 1 1 B 1 = (ξ B) (16) µ J 1 = (ξ B) 1 = ξ γ ξ where ξ is the lasma dislacement vector and quantities with a 1 subscrit reresent first order erturbations [Freidberg 214]. For resistive wall modes the inertial effects can be neglected because the corresonding growth rates are very slow comared to the characteristic MHD time a/ V Ti where V Ti is the ion thermal seed. Thus, the lasma behavior is described by F(ξ) = (17) 1

11 In other words, referring to the general ideal MHD formulation of the energy rincile [Freidberg 214], only δw(ξ,ξ) is needed to describe the lasma behavior, and ω 2 K(ξ,ξ) can be ignored. Next, form the total energy integral δw for n = modes in the usual way: δw = 1 2 ξ F(ξ)dr = (18) Using standard analysis [21] we can rewrite Eq. (18) as follows δw(ξ,ξ) = δw F (ξ,ξ)+ BT = δw F (ξ,ξ) = 1 2µ [(Q) 2 + γµ ( ξ) 2 µ ξ J Q + µ (ξ )( ξ )]dr (19) BT = 1 2µ (n ξ )[B Q γµ ξ µ ξ ]ds P S P where Q = (ξ B), n is the outward unit surface normal vector, and the surface integral in BT is evaluated on the lasma surface S given by Ψ =. This form is P general, and in articular is valid for n = modes in a tokamak with arbitrary rofiles. Many researchers have long believed that n = stability is closely related to the roblem of neighboring equilibria of the Grad-Shafranov equation. This is indeed a correct belief although to the authors knowledge a derivation of this connection has not aeared in the literature. We have derived a relationshi which exlicitly shows this connection. The analysis is somewhat lengthy and is given in Aendix A. The final result is a simlified form of δw F valid for u-down symmetric tokamaks undergoing vertical-like dislacements; that is, dislacements for which ξ Z (R,Z) = ξ Z (R, Z) and ξ R (R,Z) = ξ R (R, Z), (e.g. ξ Z = ξ = constant, ξ R = ). Because of u-down symmetry these dislacements exactly decoule from the horizontal-like dislacements for which ξ Z (R,Z) = ξ Z (R,Z), ξ R (R,Z) = ξ R (R, Z). In addition, it is shown in Aendix A that the most unstable dislacements are incomressible: ξ =. The simlified form of δw F is conveniently exressed in terms of the erturbed lasma flux ψ which is related to the dislacement vector by ψ = ξ Ψ (2) 11

12 The resulting form of δw F, valid for arbitrary equilibrium rofiles, is given by δw F = 1 ( ψ) 2 µ 2µ R R F 2 2 ψ2 dr + 1 µ J φ 2 ψ 2 R 2 B ds (21) P Where, as before, a rime denotes a derivative with resect to the oloidal flux function, i.e.d /dψ. Note that δw F is already written in self adjoint form. The connection to Grad-Shafranov neighboring equilibria is now aarent. The differential equation in ψ obtained by setting the variation in δw F to zero is identical to the neighboring equilibrium equation obtained by letting Ψ Ψ + ψ in the Grad- Shafranov equation and setting the first order contribution to zero. Observe that there is a boundary term arising from several integrations by arts in the derivation. This term is often zero since J φ (S) = on the lasma surface for many realistic rofiles. However, it is not zero for the Solov'ev rofiles since the edge current density is finite and is, in fact, is the main drive for vertical instabilities. To comensate this difficulty note that = F 2 = for the Solov'ev rofiles imlying that the volume contribution to δw F reduces to that of a vacuum field. In other words, the erturbed toroidal current density in the lasma is zero for the Solov'ev rofiles. This simlification greatly outweighs the difficulty of maintaining the surface contribution. Secifically, it will ultimately allow us to make use of an analytic form of the Green s function for the lasma region when evaluating d, a mathematical advantage that does not aly to general rofiles. The task now is to evaluate and simlify the boundary term BT, recasting it in a form that is automatically self-adjoint. The inner vacuum region The first ste in the simlification of BT in Eq. (19) focuses on the inner vacuum region between the lasma and the resistive wall. By making use of incomressibility and rewriting Eq. (2) as n ξ = ψ / RB where n is the unit normal vector to the W F lasma surface, we see that the boundary term reduces to BT = 1 ψ 2µ (B Q µ RB ξ )ds P (22) S P Next, assume that the erturbed magnetic field in the inner vacuum region is also written in terms of a flux function ˆψi. This is convenient because the revious analysis 12

13 for the lasma region in terms of ψ is directly alicable to the inner vacuum region by simly setting the equilibrium J = = in the vacuum region. Now, the jum conditions across and no surface currents) [21] S require that (for incomressible dislacements P ˆψ i SP = ψ SP ˆB ˆB i SP = (B Q µ ξ ) SP (23) with ˆB the equilibrium vacuum field in the inner vacuum region. At this oint the first natural boundary condition is introduced into the formulation by substituting Eq. (23) into Eq. (22) BT = 1 ˆψi 2µ (ˆB ˆB RB i )ds (24) S P Continuing, in the inner vacuum region the total (i.e. equilibrium lus erturbation) magnetic field can be written as ˆB Tot = 1 R (ˆΨ + ˆψ i ) e φ + ˆF(ˆΨ + ˆψ i ) e R φ (25) Since ˆF(ˆΨ + ˆψi ) = R B = constant in a vacuum region it follows that the erturbation ˆF i = ( ˆF / ˆΨ)ˆψ i =. Thus the erturbed magnetic field in the inner vacuum region for an n = erturbation is given by ˆB i = 1 R ˆψ i e φ (26) from which it follows that ˆB ˆB i SP = 1 R 2 ( Ψ ˆψ i ) S P = B R (n ˆψ i ) S P (27) 13

14 Here we have used the equilibrium continuity relation ˆB(S P ) = B(S P ) which is valid when no surface currents are resent. The boundary term can thus be written as BT = 1 1 ˆψi 2µ (n ˆψ i )ds P (28) S P R 2 The last art of this first ste is to recognize that the magnetic energy in the inner vacuum region (with subscrit i) can be written as δw VI = 1 2µ (ˆB i ) 2 dr = 1 ( ˆψ i ) 2 2µ dr (29) R 2 with ˆψi satisfying Δ * ˆψi = (3) Using Eq. (3) we can easily convert Eq. (29) into two surface integrals, one on the lasma surface S and the other on the inner wall surface S P W, δw VI = 1 ( ˆψ i ) 2 2µ dr = 1 1 ˆψi R 2 2 (n ˆψ R 2 i )ds W 1 1 ˆψi 2 (n ˆψ R 2 i )ds P (31) S W S P Here and below n always refers to an outward ointing normal. This relation allows us to write Eq. (28) as BT = δw VI 1 1 ˆψi 2 (n ˆψ R 2 i )ds W (32) S W The outer vacuum region The boundary term can be further simlified by introducing the magnetic energy in the outer vacuum region. In analogy with Eqs. (29) - (31) we can write the outer vacuum energy (with subscrit o) as 14

15 δw VO = 1 2µ (ˆB o ) 2 dr = 1 ( ˆψ o ) 2 2µ R 2 dr = 1 1 ˆψo 2µ (n ˆψ o )ds 1 1 2µ R 2 S S W R 2 ˆψo (n ˆψ o )ds W (33) The contribution at the surface at infinity, S, vanishes because of the regularity boundary condition far from the lasma. Thus, Eq. (33) reduces to δw VO + 1 2µ 1 This exression is now added to Eq. (32) leading to ˆψo (n ˆψ R 2 o )ds W = (34) S W BT = δw VI + δw VO + 1 2µ 1 R 2 [ ˆψ o (n ˆψ o ) ˆψ i (n ˆψ i )]ds W S W (35) In Eq. (35) it is understood that the ˆψi terms are evaluated on the inner surface of the wall while the ˆψ o terms are evaluated on the outer surface. Equation (35) is in a convenient form since in the limit of a thin wall the surface terms are ultimately transformed into a simle set of jum conditions arising from the solution for the fields within the resistive wall. The fields within the resistive wall The erturbed fields within the resistive wall are determined as follows. We write B w = A w E w = γa w V w = γa w (36) J w = σe w Here, σ is the wall conductivity, the scalar electric otential V has been set to zero as w the gauge condition, and all quantities are assume to vary as Q(r,t) =Q(r)ex(γt). Amere s Law then can be written as A w = µ σγa w (37) 15

16 Now, form the dot roduct of Eq. (37) with e φ / R and define A w e φ = ψ w / R. A short calculation for n = symmetry yields 2 ψ w 2 R ψ w R = µ σγψ w (38) It is at this oint that the thin wall aroximation is introduced in order to obtain an analytic solution for ψ w. Assume that the wall thickness is denoted by d while the minor radius of the wall at Z = is denoted by b L W / 2π where L W is the wall circumference. The thin wall aroximation assumes that d /b 1. The thin wall ordering is then given by µ γσbd γτ w 1 n 1 d (39) t 1 b The hysical interretation is as follows. The growth time γ 1 µ σbd τ w is the characteristic diffusion time of magnetic flux through a wall of thickness d and conductivity σ into a vacuum region of thickness b. The unit vector n is the outward normal to the wall and the ordering n 1 /d imlies raid variation across the wall. Similarly the unit vector t is tangential to the wall in the oloidal direction. The ordering t 1 /b imlies that tangential variation is slower (on the scale of the device size) than normal variation. The analytic solution to Eq. (38) can now be obtained using the resistive wall analog of the constant ψ aroximation that arises in tearing mode theory [Furth et al. 1963]. We define a tangential oloidal arc length l and a normal distance s measured with resect to the inner surface of the resistive wall such that < s < d. This imlies that n = s (4) t = l 16

17 The thin wall exansion for ψ w is then written as ψ w (s,l) = ψ(l)+! ψ(s,l)+ (41) where ψ! / ψ d /b. After some analysis we find that the first non-vanishing contribution to Eq. (38) is given by 2 ψ! s = µ σγψ (42) 2 This equation can be easily integrated leading to the following analytic solution for ψ w ψ w (s,l) = 1 + µ σγ s 2 2 +c +c s 1 2 d ψ (43) where c 1 (), l c 2 () l are two free, order d / b integration constants arising from the solution to Eq. (42). The critical information to extract from these solutions is the change in ψ w and its normal derivative across the resistive wall. These values are given by ψ w (d,l) ψ w (,l) = ψ αd 2 +c 2 d b ψ n ψ w (d,l) n ψ w (,l) = ψ (d,l) w ψ (,l) w = αψ s s (44) Here, α = γµ σd 1 /b is related to the resistive wall growth time. Observe that there is a small, negligible jum in the flux across the wall. There is, however, a finite jum in the normal derivative, reresenting the current flowing in the wall. The information in Eq. (44) is related to the inner and outer vacuum solutions by recognizing that even though the wall is thin, it still has a small finite thickness. The imlication of finite thickness is that there are no ideal infinitesimally thin surface currents on either face of the wall. Therefore, across each face of the wall the flux and its normal derivative must be continuous with the adjacent vacuum fields. Secifically, couling from the wall to the vacuum regions requires that 17

18 Inner Surface Outer surface ˆψ i (s,l) ψ w (s,l) = s= ˆψo (s,l) ψ w (s,l) = s=d n ˆψ i (s,l) n ψ w (s,l) = s= n ˆψ o (s,l) n ψ w (s,l) = s=d (45) By combining Eqs. (44) and (45) we obtain a set of jum conditions on the vacuum magnetic fields that takes into account the effects of the resistive wall. Simle substitution yields ˆψ o (d,l) ˆψ i (,l) = n ˆψ o (d,l) n ˆψ i (,l) = α ˆψ i (,l) (46) This is the information required to comlete the variational rincile. The final variational rincile Return now to the exression for the boundary term given by Eq. (35). By substituting the to relation in Eq. (46) we obtain an exression for the boundary term in the thin wall limit that can be written as BT = δw VI + δw VO + 1 2µ ˆψi R 2 [(n ˆψ o ) (n ˆψ i )]ds W S W (47) The second natural boundary condition is introduced into Eq. (47) by substituting the bottom relation from Eq. (46), BT = δw VI + δw VO + α 2 ˆψi 2µ R ds (48) 2 W S W Observe that the last term is now in a self-adjoint form. By combining Eq. (48) with Eq. (19) we finally obtain the desired variational rincile 18

19 δw = δw F + δw VI + δw V + αw D = δw F = 1 2µ ( ψ) 2 µ R R F 2 2 ψ2 dr + 1 µ J φ 2µ ψ 2 ds R 2 B P V P S P δw VI = 1 ( ˆψ i ) 2 2µ dr V I R 2 δw VO = 1 ( ˆψ o ) 2 2µ dr V O R 2 (49) W D = 1 2 ˆψi 2µ R ds 2 W S W The variational rincile is very similar in form to ideal MHD. When substituting trial functions all that is necessary is to insure that the erturbed fluxes are continuous across the lasma-vacuum interface and across the resistive wall. The normal derivative requirements are automatically accounted for by means of the natural boundary conditions. 4. Summary We have resented a general formulation of the n = resistive wall stability roblem. The end result is a variational rincile which, as shown in the accomanying aer, is quite amenable to numerical analysis. The key new feature introduced in the analysis is the resence of a feedback system. A simle but reliable measure of the effect of feedback is determined by calculating the maximum stable k and corresonding d for fixed γτ w αb 1. Numerical results obtained with this formalism and imlications for reactor designs are given in the accomanying aer. Acknowledgments The authors would like to thank Prof. Dennis Whyte (MIT) for roviding the motivation for this work and for many insightful conversations. J.P. Lee and A.J. Cerfon were suorted by the U.S. Deartment of Energy, Office of Science, Fusion Energy Sciences under Award Numbers. DE-FG2-86ER53223 and DE-SC J. 1 The recise definition of the wall diffusion time, τ w = µ σdl W / 2π with L the wall W circumference, will aear naturally during the discussion of the numerical analysis. 19

20 P. Freidberg was artially suorted by the U.S. Deartment of Energy, Office of Science, Fusion Energy Sciences under Award Number DE-FG2-91ER

21 1. Introduction Aendix A Relation between MHD stability and neighboring equilibria We resent a derivation of the fluid contribution ( δw F ) to the total otential energy ( δw ) describing the n = stability of a tokamak. The derivation is resented for arbitrary rofiles and then simlified at the end for the Solov ev rofiles. The derivation requires a substantial amount of analysis. However, the final result is quite simle and is shown to be directly related to neighboring equilibria of the Grad- Shafranov equation. 2. The starting oint The starting oint for the analysis is the exression for the total MHD otential energy δw written in the standard form (see Eq. 19) [21] δw(ξ,ξ) = δw F (ξ,ξ)+ BT δw F (ξ,ξ) = 1 2 [(Q) 2 + γ( ξ) 2 ξ J Q +(ξ )( ξ )]dr (A.1) BT = 1 2 S P (n ξ )[B Q γ ξ ξ ]ds P Here Q = (ξ B) and for convenience µ has been set to unity. It can be reinserted at the end of the calculation. The quantity δw F is the fluid contribution while BT is the boundary term that will ultimately be related to the vacuum energy and the resistive dissiated wall ower. Note also that for the n = mode in an u-down symmetric tokamak we can assume that ξ is urely real. The resent analysis is focused solely on obtaining a simle form for δw F. Observe that even though our goal is to calculate the eigenvalue ω for resistive wall modes, we can neglect the ω 2 contribution due to the lasma inertia. The reason is that resistive wall growth rates are much slower than ideal MHD growth rates, thus justifying the neglect of MHD inertial effects. This is the exlanation of why only δw(ξ,ξ) and not ω 2 K(ξ,ξ) is needed to describe the lasma behavior. The eigenvalue ω will aear in the evaluation of BT. 3. Incomressibility 21

22 The first ste in the analysis is to examine the lasma comressibility term. As is well known this is the only term in which ξ! aears. The question is whether or not a well behaved ξ! can be found that makes ξ = which, if ossible, clearly minimizes the lasma comressibility term. To answer this question we have to make use of the following symmetries with resect to the Z deendence of the equilibrium magnetic fields B R (R, Z) = B R (R,Z) B Z (R, Z) = +B Z (R,Z) (A.2) B φ (R, Z) = +B φ (R,Z) Now, for the n = mode a general erturbation ξ can be always be written as the sum of an even ξ Z contribution lus an odd ξ Z contribution which are comletely decouled from one another because of the equilibrium symmetry. The erturbation symmetries are as follows, Even ξ Z Symmetry Odd ξ Z Symmetry ξ Z (R, Z) = +ξ Z (R,Z) ξ R (R, Z) = ξ R (R,Z) ξ φ (R, Z) = +ξ φ (R,Z) ξ! (R, Z) = +ξ! (R,Z) ξ (R, Z) = ξ (R, Z) ξ Z (R, Z) = ξ Z (R,Z) ξ R (R, Z) = +ξ R (R,Z) ξ φ (R, Z) = ξ φ (R,Z) ξ! (R, Z) = ξ! (R,Z) ξ (R, Z) = + ξ (R, Z) (A.3) The n = modes of interest have even ξ Z symmetry (e.g. x = constant ) corresonding to vertical dislacements. The less interesting modes have odd ξ Z symmetry (e.g. ξ R = constant ) and reresent horizontal dislacements. For the vertical dislacements of interest the most unstable modes are always incomressible. To show this we note that for a minimizing incomressible dislacement ξ! must satisfy Z 22

23 B ξ! B = B ξ! B = B l ξ! B = ξ (A.4) where B is the oloidal magnetic field and l is oloidal arc length. Thus, for ξ! to be well behaved (i.e. be eriodic) it must satisfy the eriodicity constraint = ξ! " dl = ξ l B " dl = ξ B " dz (A.5) B Z This constraint is automatically satisfied for the even ξ Z symmetry dislacement. The conclusion is that the lasma comressibility term in δw F is minimized by choosing ξ! to satisfy Eq. (A.4) leading to the result γ( ξ) 2 =. The remainder of δw F is only a function of ξ. 4. Reformulating δw F in terms of the vector otential The evaluation of δw F is more conveniently carried out in terms of the erturbed vector otential A 1 rather than the lasma dislacement ξ. The familiar relation between these two quantities is A 1 = ξ B ξ = B A 1 B 2 (A.6) In general A 1 can be vector decomosed as follows A 1 = f (R,Z) Ψ + g(r,z)b + h(r,z)e φ (A.7) Here, Ψ(R,Z) is the equilibrium flux function satisfying the Grad-Shafranov equation. We next make use of the constraint A 1 B = and introduce the erturbed flux ψ(r,z) = RA φ1 (R,Z) = Rh(R,Z). A short calculation then enables us to rewrite A 1 as A 1 = f Ψ Fψ R 2 B 2 B + ψ R e φ (A.8) where F(ψ) = RB φ. The basic unknowns in the roblem are ψ, f. 23

24 Before roceeding with the evaluation of δw F it is useful to list a number of relations involving A 1 that enter the analysis. These are relatively straightforward to derive. Q = A 1 = 1 R ψ e + f Ψ + 1 Ψ Fψ φ R R 2 2 B FJ ψ φ R 2 2 B e φ Q A 1 = ψ 1 Ψ Fψ R R R 2 2 B FJ ψ φ R 2 2 B Fψ Ψ ψ R 4 2 B + 2ψ(B f ) (ψf B ) Q B = 1 R Ψ ψ + F 1 Ψ Fψ 2 R R R 2 2 B FJ ψ φ R 2 2 B + F(B f ) ψ (e φ A 1 ) = Fψ R 3 B 2 Ψ ψ Rf B ψ (A.9) ξ = ψ A 1 ( Ψ B) = B 2 ψ with rime denoting d /dψ as in the main text. We are now ready to evaluate δw F 5. Evaluation of δw F There are three terms aearing in δw F as given by Eq. (A.1) after setting ξ =. Using Eq. (A.9) these terms can be evaluated in a straightforward way. The magnetic energy term is given by (Q) 2 = ( ψ)2 + 1 Fψ Ψ R 2 R R 2 2 B FJ ψ φ R 2 2 B +(RB f ) 2 + 2(RB f ) 1 Ψ Fψ R R 2 2 B FJ ψ φ R 2 2 B 2 (A.1) 24

25 The current term is evaluated by making use of the equilibrium relation J = F B + R e φ which leads to a fairly comlicated exression ξ (J Q) = F + F B 2 (A Q)+ 1 B ψ(b Q) 2 = F + F ψ 1 Ψ Fψ B 2 R R R 2 2 B FJ ψ φ R 2 2 B Fψ Ψ ψ R 4 2 B +2 ψ R (RB f ) (ψf B ) + ψ 1 B 2 R Ψ ψ + F 1 Ψ Fψ 2 R R R 2 2 B FJ ψ φ R 2 2 B + F R (RB f ) The remaining ressure term requires some minor algebraic maniulations. The result is (ξ )( ξ ) = [(ξ )ξ ] ξ (ξ ) (A.11) = ( ψξ )+ ψ B [A ( Ψ B)]+ 2 1 B [A ( ψ B)] 2 1 = ( ψξ ) ψ 2 + F RB [ ψ (e A )] ψ ( Ψ ψ) 2 φ 1 R 2 2 B = ( ψξ ) ψ 2 ψ R 2 2 B F ( Ψ ψ) RB [f (RB ψ)] 2 (A.12) The next task is to sum these three comlicated contributions and simlify the result. The terms in the sum can be collected and written as follows δw F (ξ,ξ) = 1 2 [(Q) 2 ξ J Q +(ξ )( ξ )]dr = 1 2 U dr U =U 1 (f, f )+U 2 (f, ψ)+u 3 (ψ, ψ) (A.13) Here, each U is a combination of algebraic and differential oerators acting on the j terms aearing in the arguments. After a short calculation the first two terms on the right hand side can be simlified leading to 25

26 U 1 (f, f )+U 2 (f, ψ) = (RB f ) 2 + ( F f ψb ) +2(RB f ) 1 Ψ Fψ R R 2 2 B FJ ψ φ R 2 2 B F ψ R (A.14) The divergence term integrates to zero over the lasma surface. The f deendence of the remaining terms involves only (RB f ). These terms are simlified by comleting the square on (RB f ). The fluid energy δw F is then minimized by choosing (RB f ) = 1 Ψ Fψ R R 2 2 B + FJ ψ φ R 2 B + F ψ 2 R (A.15) Note that as for the incomressibility analysis, the eriodicity constraint on (RB f ) is automatically satisfied for the erturbation with even ξ Z symmetry. The end result is that U 1 (f, f )+U 2 (f, ψ) = 1 Ψ Fψ R R 2 2 B FJ ψ φ R 2 2 B F ψ R 2 (A.16) which is only a function of ψ. The comlete integrand U is now evaluated by adding Eq. (A.16) to U 3 (ψ, ψ). After another tedious calculation we obtain U = ( ψ)2 F + 2 R 2 R + F F J φ 2 R 3 2 B ψ2 ( ψξ ) F + ψ Ψ Fψ R 2 R 2 2 B + F ( Ψ ψ) R 2 2 B (A.17) After some further maniulations that make use of the equilibrium Grad-Shafranov equation we find that the term on the second line of Eq. (A.17) can be rewritten as F ψ Ψ Fψ R 2 R 2 2 B + F ( Ψ ψ) R 2 2 B = F F ψ 2 R 4 2 B Ψ F F F F J φ R 2 R 3 2 B ψ2 (A.18) 26

27 Equation (A.18) is substituted into Eq. (A.17). The divergence terms are converted into surface integrals and then simlified using the equilibrium Grad-Shafranov equation. This leads to the final desired form of δw F δw F = 1 ( ψ) R 2 2R F 2 2 ψ2 dr + 1 J φ 2 ψ 2 R 2 B ds (A.19) There are three oints worth noting. First, a simle variational analysis shows that the function ψ that minimizes the volume contribution satisfies S P Δ * ψ = R F 2 ψ (A.2) This equation is identical to the erturbed Grad-Shafranov equation corresonding to neighboring equilibria. Second, the surface contribution is nonzero only if there is a jum in the edge current density. For many rofiles J φ is zero at the lasma edge but it is finite for the Solov ev rofiles. Third, there is a great simlification in the volume contribution for the Solov ev model in that = F 2 =. Thus the fluid energy for the Solov ev model reduces to with ψ satisfying δw F = 1 ( ψ) 2 2 R 2 dr + 1 J φ 2 ψ 2 R 2 B ds (A.21) S P Δ * ψ = (A.22) Assuming a solution to this equation can be found then the volume contribution can be converted into a surface integral. The fluid energy can then be written solely in terms of a surface integral δw F = R ψ n ψ + J φ ψ 2 2 R 2 B ds (A.23) S P 27

28 This form is actually valid for any equilibrium rofile. The beauty of the Solov ev rofiles is that n ψ on the lasma surface can be conveniently evaluated in terms of ψ on the surface using Green s theorem since we know the analytic form of the Green s function corresonding to Eq. (A.22). For general rofiles this rocedure is not as convenient since we do not know the Green s function in a simle analytic form. 28

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32 Figure Cations Paer 1 Z max /R δϵ Z/R ϵ κϵ Z min /R R min /R 1 R max /R R/R Figure 1 Geometry of the lasma equilibrium Z/R R/R Figure 2 Flux surfaces for NSTX using the Solov'ev model. The value of A has been chosen such that J φ (1 ε,) = 32