LA-9055-MS
UC-2O9 LA—9055-HS Issued: January DE82 010020
Comments on Ideal Ballooning
R. Y. Dagazian R. B. Paris*
* Visiting Staff Member. Centre d'Etudes Nucleates de Fontenay-aux-Roses, FRANCE.
Los Alamos National Laboratory Los Alamos, New Mexico 87545
T'OV CF ••"•' •'' ••.-.•.,• .<; (ii'imiTn COMMENTS ON IDEAL BALLOONI"C
by
R. Y. Dagazian and R. 5. Paris
ABSTRACT
Ideal ballooning modes are investigated for the case of plane magnetized slab geometry. Toroidal effects are simulated by a gravitational acceleration periodically varying along magnetic field lines. High shear is shown to be very effective in stabilizing these modes even when field line curvature is most unfavorable to their stability.
I. INTRODUCTION
Ballooning modes can impose a serious limit on the amount of plasma that can be stably confined in a toroidal magnetic confinement device. The infinite conductivity, ideal, ballooning mode has been extensively studied in the literature for realistic plasma confinement configurations of the tokamak type.1"3 All of these studies, to our knowledge, neglect the effects of compressibility and of strong shear. It Is the purpose of this report to estimate the importance of these effects by employing the simple model of the magnetized plasma slab subject to a modulated gravity simulating varying toroidal curvature along magnetic field lines. We find that, in contrast to the real toroidal case, ideal ballooning modes in the plane slab are necessarily incompressible in the limit of large perpendicular wave numbers. On the other hand, strong magnetic shear is very effective in stabilizing these modes, even in the most adverse situations. Devices like reversed field pinches (RFP) can be stable for rather high values of kinetic to magnetic pressure ratio (beta). II. THS MODEL We consider the case of the magnetized plasma slab equilibrium with magnetic shear. We write the equilibrium magnetic field as
B(o) = [0, Byo)(x), B where x, y, and z are Cartesian coordinates and no a priori assumption is made as to the relative magnitude of the two field components. Toroidal curvature effects are simulated by introduction of a gravitational acceleration modulated along a given line of force GQ [-0, + cos^2]ex , Go = 30 J- . (2) Hare, Rc is the simulated radius of curvature of the magnetic field, a is a characteristic dimension of the plasma, and L is the connection length along the magnetic field lines. The parameter a simulates average curvature along these lines. Taking the ex • VxVx and the i$ = B /|g^ '| components of the equation of motion together with the induction and continuity equations, we combine the normalized system of the linearized set of magnetohydrodynamic (MHD) equations, and the adiabatic state law into the form p(0)w[..72v(l) + 3 V.V(D] = _F(0)72B(l) + F(0) B(l) _ 2r_3_ + ^ijp(l)G(z) (3) (4) (O;) (1 F vv > - 0 , (5) and Here symbols have their usual meanings and we have employed Che normalization v r a P B — + v , -_ + t , — + p , — + P , — + B vh ~ Th PO p0 B0 ~ with 0 B0 F<0) SB(O)_L+B^>2- , andF<°> , ^L_ 1. y 3y z 8z dx2 9y BQ, PQ, and PQ represent characteristic values, all distances are normalized to a and y in the adiabatic index. The II subscript indicates the direction parallel to the equilibrium magnetic field, and the time dependence of modes is taken of the form exp(wt). Finally, in deriving Eq. (3), constant mass density has been assumed. We now consider ballooning modes of the twisted slicing form"',^ f(1)(x,y,z,t) = f(6x,z) exp [itc(B^0)z - B^0)y) + oat] , (7) where <5 << 1 is an equilibrium-profile-dependent parameter and k is the wave number perpendicular to the field line 0) B< z - B(°)y = co(x) . (8) Neglecting the variation of perturbed quantities on the scale of the equilibrium profile variation (6 •+ 0), we combine Eqs. (3) and (4) with Eq. (5) in the limit k •> °°: a v(i) _ u2p<0) (1 + S2z2) v(i) + 2G(Z) (i) „ 0 3z x x and Here dB(0) dB(0) y Z - BC0) focusing on a field line with CQ(X) =0.'* Furthermore, taking the divergence of the equation of motion we find in the same limit representing constant total pressure to leading order in k2. From the flux-freezing condition, we have1* 2 V . v^> =^V(O _uB(l)/B<0) , ^ -Ii- lnB(") . (12) Combining Eqs. (6), (10), (11), and (12) it is easy to show that (13) The trivial solution of Eq. (13) is the only physical solution; hence the modes we are concerned with are indeed incompressible in the large k2 limit. In this limit, Eq. (9) describing ideal ballooning becomes simply ^0)2 _L(1 + s2z2) JL v - o.2(l + s2z2) vd) + 2yrG(z)v<1) = 0 . (14) z 3z 3z x x x Here we have normalized pressure, mass density, and magnetic fields to their local values, p(0) = p(0) = B^°) = 1. The result derived in Eq. (13) does not hold in general toroidal geometry. Adopting the usual general toroidal coordinates (tp, x> ?) such that -vp(0) vp and following the same procedure as above, we can show that C15) and 1) . ,,(1) = -0V . Y(D _ 2 where B^0) 3h<, B<0) 3h, -I = - ——.— i^— -K— - K-T^I Ce „ • V) In 1 B«»h<% JI "X and witli the h^ denoting the metric coefficients. The term KT^ is the magnetic curvature of the configuration in the pressure gradient direction, while <, is a curvature-related coefficient. Both Kj and <^ vanish in the plane geometry case. Combining Eqs. (15) and (16), we deriv a) [4^)2 J2] «» ^0) __ The gravitational acceleration G was introduced in Eq. (3) exactly to simulate the effects of K^ in toroidal geometry. Although G does not occur in Eq. (13) derived for plane geometry, IO does enCer the toroidal geometry expression, Eq. (17). The right-hand side of Eq. (17) is negligible for the case of a tokamak, making a contribution of O(6g/A ) to the equation of motion. However, this is not the case for an RFP because B^ "*/B^ can now be of 0(1). For the case of zero shear, Eq. (14) can be put into the form of the Mathiei1 equation. Defining 2 2 2 a B -4R (u> + 2oG0) , q = 4R GQ , GQ = YeOra/Rc , (18) and (0) R = L/2TTBZ we can find simple analytic results for the limits 2 2 q « 1: a) ~ 2GQ R - 2aGQ (19) and 2 q >> 1: UJ ~ 2 G0(l - a) - Go /R . (20) When shear is finite, the disconnected mode approximation can be applied. By letting z = sz, Eq. (14) can now be put into the form 2 2 2 A (1 + z ) _!_ vv - [X + Y (1 + z )] vv = 0 , (21) 3z 8z where C - 2G0 r, 1 = " 7%^ Ll " °+ 2 and Equation (21) is the zeroth-order oblate spheroidal equation. A discussion of its eigenvalues for solutions decaying in the limit |z| + B is given in the Appendix. Here we merely state the principal results. There are two limits for which analytic estimates for the eigenvalues can be obtained: (a) y << !• We find from Eq. (A-17) the largest even eigenvalue where 1/2 u E (- U - 1) , For u + 0, 4>(u) can be expanded to find from Eq. (A-16) Y2 ~ C2e"4ir/>u, C = 16e1T/2"Y with Y = 0.5772... being Euler's constant, so that 2 2TTSB [^L] + C exp( 1 , ) . (23) 2 2 2J {2O[la + I@ lI B^}" (b) Y2 >> 1. In this case we find from Eq. (A-18) -X = Y2 + Y + - + 0(Y~J) 2 or 1/2 2 = 2G U - a) - sB (0) {2G - a + — (24) 0 f 2 sL' Numerical solution of Eq. (14) for the eigenvalue <^2 poses no particular difficulty for the case of finite shear. For a treatment of this problem we followed essentially the procedure described in Ref. 4. It is found that the convergence of the iteration scheme is rapid for peaked modes but becomes slow as the shear parameter s approaches zero. However, for s = 0, exact solutions of Eq. (14) may be obtained from Mathieu-equation theory. III. RESULTS AND DISCUSSION The salient results of the present report are summarized in Fig. 1. The critical Gg beyond which instability arises has been numerically evaluated as a function of sL for a wide range of parameters obtainable in experiments. We have chosen R = 5 as a representative value for any actual toroidal device. For tokamaks, B (0) " 2 is of 0(1); L is OL the order of the aspect ratio A and R ~ Aq, where q is the safety factor. In an RFP, L is in general shorter but B is only a fraction of B . Positive values of a correspond to tokamaks, spheromaks, and possibly the near-axis regions of RFPs. We remind the reader Fig. 1. The critical GQ as a function of shear with a as the parameter. A possible set of conditions over the poloidal section of an RFP is indicated by a dashed line. that this quantity is related to V"(i|/). Negative values of a correspond to the region near the field-null in spheromaks and RFPs; average field line curvature is unfavorable to stability in such regions. It is seen that sizeable values for the critical SQ can be obtained provided that the product sL is sufficiently high. However, in the limit of zero shear, it is seen that Gcr|L is nonzero only for positive values of a. The dashed curve shows a possible set of equilibrium conditions as one moves from the inside toward the outside of an RFP. This device does exhibit a much larger variation in physical parameters as such a trajectory is followed. Letting yV = 1 corresponds to setting a = r the pressure gradient radius. Then GQ(CI, r , s, L) is just 60r /Rc. We note that both r and Rc can be large near the magnetic axis of a toroidal device. For the tokamak, R remains approximately constant throughout the discharge cross section and is roughly equal to the torus major radius RQ. For the RFP this quantity varies from the major radius on axis to the minor radius at field reversal. It is difficult to r specify the ratio p/Rc without a detailed equilibrium model. However, considering a pressure distribution of the form exp (-r^/r_2), we can have r or a or l e roughly Gcri(. ~ 60 p/Ro ^ Lokamak ^ near-axis region of an RFP, and r 'criL ~ 8Q for the field reversal region of the latter device. Figure 2 shows the dependence of co2 on shear with GQ as the parameter (the connection length is kept constant because we fix R). It is seen that no matter how high G (and we have indeed allowed very large values of this quantity), a value of sL can be found beyond which the plasma is stable against Ideal ballooning. Equations (22) and (23) are fairly accurate In predicting the cutoffs and the general features of the curves for the RFP-relevant set of parameters. However, Eq. (24) can accurately account only for the results shown in Fig. 2(b) involving high GQ values. Although the parameters relevant to lower 6Q values in the tokamak case, Fig. 2(c), do involve large y2. the low values of sL lead to a breakdown of the disconnected mode approximation, which has been employed in deriving Eq. (21). As we let L •*• °°, Eq. (23), unlike Eq. (2A), remains valid and yields the familiar plane Suydam criterion for g-mode stability D E 2GQ (1 - a)/s^ i\"> - JL < 0 , (25) 10 Fig. 2. The dependence of the ideal ballooning growth rate on shear with Go as the parameter. Favorable average field line curvature: (a) low Go; (b) high GQ; (c) unfavorable line curvature with high shear. evaluated at the worst curvature point on a given field line. Figure 3 shows the marginal stability curve representing the generalization of the plane Suydam Fig. 3. Marginal stability diagram for various sets of parameters. For a given large R he numerical curves corresponding to different a are indistinguishable* Observe that the disconnected mode approximation derived in the limit Y2(OJ2 = 0) >> 1 holds remarkably well even for small values of this quantity. 1 1 criterion to include ballooning. For y2(u)2 =0) >> 1, an analytic expression can be obtained from Eq. (24) in the form ;2((U2 = 0) = [D - A] (26) 4 2 2 When 7 (u = 0) « 1, we have to solve a Lranscendental equation (cf. Eq . (22)). Figure 4 shows the growth rate as a function of GQ with sL as the parameter for (a) a tokamak and (b) an RFP-relevant case. 0 Figure 5 shows the dependence of GcriL on the parameter R = L/2TTB£ \ Note that this quantity represents an effective connection length. The vanishing of B^0' then corresponds to R •+<*>, which is precisely the g-modc limit of the present theory. This is the limit relevant to RFP stability neir the field reversal surface. Modes then tend to be g-mode-like rather than ballooning-like on this surface. The critical GQ for instability is greatly enhanced for a modest reduction in R; the growth rate decreases correspondingly. Finally, Fig. 6 shows some examples of typical eigenfunctions for various discharge parameters. It is seen that for situations with high enough sheaf our modes tend to be highly peaked, thus justifying our disconnected mode approximation. When shear is low, the modes are broader about a given bad curvature point on the magnetic field line, but thsy exhibit undulations peaked on successive bad curvature points as infinity is approached. 0.50- 0.25 - Fig. 4. Ideal ballooning growth rate as a function of Go with shear as the parameter: (a) a tokamak and (b) an RFP-relevant case. 12 Fig. 5. The gritical Go for idea'l ballooning instability as a function of shear with a and R as the parameters. I 2 Typical ideal ballooning eigenfunctionsx (a) a = .05, sL = 0.5, Go = .5 x 10" , 2 3 2 1 cu = .13 x 10" ; (b) a = -1.5, sL = 25, GQ = .10, M = .21 x 10" . In conclusion, we have studied ideal ballooning modes simulating toroidal effects by means of a modulated gravitational acceleration along magnetic field lines in plane slab geometry with strong magnetic shear. It was shown that these modes are incompressible for the cases of plane geometry and for tokamaks, whereas strong curvature terms result in V • v t 0 for an RFP-like configuration. From our results, it seems that even for the most severe choice of parameters for an RFP, this device can be easily stable against ideal ballooning. Shear is very effective in stabilizing these modes, and it works even when magnetic field line curvature is most unfavorable. The tokamak relies 13 mainly on favorable curvature to stabilize Ideal ballooning; this results in general in lower 0crlt limits for this device than for an RFP. APPENDIX EIGENVALUES OF THE OBLATE SPHEROIDAL EOUATION I. BASIC PROPERTIES OF THE OBLATE SPHEROIDAL FUNCTIONS We present the details of the calculation of the eigenvalues of the oblate spheroidal differential equation of order u, expressed in the canonical _! (1 - z2) £l + [X + Y2 (1 - z2) - U2(l - z2)~]Jy = 0 . (A-l) dz dz Equation (21) is then obtained from Equation (A-l) by putting z = iz and u = 0. The boundary conditions for the ballooning problem are that y(z) be an even solution with y(z) + 0 as z •>• ± °°. For given \,ix this constitutes an eigenvalue problem for the free parameter y (or conversely, for given y,\i the eigenvalue is A). Before determining the eigenvalues, we set down the necessary basic properties of the oblate spheroidal functions that we shall require. There are two limiting cases of Eq. (A-l) for which siimle solutions may be given. When |z2| » 1, Eq. (A-l) reduces to Bessel's equation with the solution where Z denotes' any Bessel function. When y2 = 0 (or when Y2lz2l « X), Eq. (A-l) reduces to Legendre's equation with the solutions P^z) and Q 14 These limiting cases suggest solutions of Eq. (A-l) in the form of expansions in terms of these functions. A first set of oblate spheroidal functions involving an expansion in Bessel functions is 2 2 11 (z;Y) = (1 - z" )"^ [A» (Y)]"]" !_ a2r *yjr(yz), j = 1,2,3,4 (A-3) where i 1/2 and U,» respectively. The coefficients a-.. = aJJ 2r(y) satisfy a three-term recurrence relation8 and are normalized such that ajj Q (0) = 1 with av,2r as Y2 •* 0. The quantity Ay (Y) is a normalizing factor chosen to be r= r so that as z + =» in |arg (YZ) | Su(j) (z;Y) „ ^(j) (yz) f j = 1>2,3,4 ; z + « in |arg (yz) | 15 The solutions In Eq. (A-3) may be shown to converge in |z| > 1. A second set of oblate spheroidal functions satisfying Eq. (A-l), which involves the Legendre functions, is given by P ( v+2r and r (z;T) = I (-) a2 Q» (z) , (A-6) r=-« v+2r where the coefficients a2r are the same as in Eq. (A-3). These solutions converge everywhere (except possibly at z - ±1 and °°). Because Eq. (A-l) is a second order linear differential equation, there are only two linearly independent solutions and the different expansions in Eqs. (A-3) and (A-6) must be related. Only the relations relevant to the solution of the ballooning problem will be given here. From Eqs. (20) and (28) of Ref. 8 (Section 16.9) we have 3 1 ! cos TTV S^ ) (z;Y) - e"™ S^D (Z;Y) - i S^ > (z;Y) , (A-7) V V -V-l (I) H and from standard properties of the Legendre functions Qsv (z;y) sin7r(v - M) = Qsp(z;y) simi(v + y) - •ne71^1 COSITV PSP (Z;Y) • (A-9) -v-l v v The quantity K^j (y) appearing in Eq. (A-8) is a constant (depending upon v, \i and y) knovm as the joining factor that links the S^l^(z;y) and Qs^ _I(Z;Y) functions. Its value may te determined by expanding both sides of Eq. (A-8) In a Laurent double series in powers of z2 and equating coefficients of like powers 16 of z2k, k = 0,±l,±2,... . Because K^ (Y) is independent of k, it is necessary to consider only the coefficients corresponding to k = 0, to find8 wvi 1 = I (1) r(l + v - u)e [A{j (Y)]" (-r.'mv + r + ,,,, _ (A_1Q) I // 2f r^0 r!T(l/2 - v - r) Combination of Eqs. (A-7) - (A-9) yields (Z;Y) = A Psu (Z;Y) + B Qsy (z;Y) V V V 'p (z) + BQM (z)j , (A-ll) v+2r v+2r where = -27rVi r^1 sin7f(V + M) [e-2"vi KM (Y) + ie^ <^ (Y)] . (A-12) TTCOSTTV v -V-l Equation (A-ll) represents the analytic continuation of S^J (Z;Y) inside the unit circle |z| < 1. Similar continuations may be obtained for the other Bessel function expansion solutions in Eq. (A-3), but these will not be required in this Appendix. This completes the summary of the properties of the solutions of Eq. (A-l) for the determination of the ballooning eigenvalues. 17 II. THE EIGENVALUE RELATION FOR y IN TERMS Of' u AND X '.tie desired solution behaving correctly at <*> is y = S^" (ix;y) for x > 1 (where x = z), because the Hankel function H^^ (ix) behaves like Ky(x) when x > 0. The continuation of this solution onto the interval 0 < x < 1 is given by Eq. (A-ll). The definition of the Legendre functions in terms of Gauss hypergeometric functions in |z2| < 1 [with Im(z) > 0 because x > 0j is u y 1/2 2 2 2 2 P (z) = 2 Tt (z - l)""^ [aF, (z ) + b?F9 (z )] v+2r L L and QM (Z) = 2% (z2 _ D-U/2 eTTni rcF (z2) + dzF v+2r where ;l; z2) and a = Tfl-Iv -ii - r) rf 1 +Iv -Ip +r' 4 2 2 ; l 2 2 -2 18 r(I + Iv + IJJ + r) i *i(n-v-l-2r)/2 2 2 2 c = 2 e T(l + Iv - Iy + r) iri(u-v-2r)/2 The conditions for an even or odd solution, given by the vanishing of the coefficients of zF2(z2) and Fj(z ) in the summand of Eq. (A-ll), are consequently Ab + Bde71^1 = 0 and Aa + Bce1^1 = 0 respectively, which become r A~ = I B(-) l.v + lu + r) r[ - Iv - LV - r) (even) v) and 1 A = - I (odd) cos— + v) Inserting the values of A and B from Eq. (A-12) then yields the conditions for Sy (ix;y) and its continuation, Eq. (A-ll), to be either an even or odd function of x, expressed in terms of the joining factor, 19 (y) = 0 , M ± v * 0,±l,±2,... , where the upper and lower signs correspond '_o the even and odd solutions respectively. Equation (A-13) represents the eigenvalue relations for the even ,nd odd eigenvalues, ye and YO» in terms of u and A (or, equivalently from E> (A-2), v). In theory, Eq. (A-13) could be solved numerically provided the coefficients are known. Such a procedure would necessarily be iterative because the coefficients a9_ = a^ (y) depend on y. z v,2r III. THE EIGENVALUE RELATION FCR y2 « 1 When y2 « 1 we can approximately solve Eq. (A-13) by making use of Eq. (A-4) to find AM (y) ^ 1, and in turn from Eq. (A-10) v (I - „) 2 and similarly for K^ (Y) with v replaced by -v-1. Substitution of this -v-1 expression for the joining factors into Eq. (A-13) and use of the reflection formula for the gamma function yields the approximate even and odd eigenvalue relations .cos i —TT ( U ~ V ) -£ = o respectively. Defining v = -•— + —lu, u = (-4A - 1)1/2 (because in the physical problem = arg F + 11 w e have A < 0), together with 6j = arg r(i + u + jlu), 92 ^ J*- )' 20 cos -ir(u ± - ~ -iu) exp i{a log - + 2 6j - 4 62} = — cos j*{» ± - + yiuj = exp i[w(u ±1) where cos = ± arctan 1 T sin wM e-*W2 with | Y(n) ~ 4 exp I 2 9- - 8. + ITTJJ - * - (n T I)ir , (A-15) p.o u z L 2 e>° 4 where n is an integer. Although the general case of finite y may be applicable to a toroidal treatment of the RFP problem (with y * 0 resulting from finite V • v), we now confine our present discussion to the case y = 0. We hope a more complete discussion of the eigenvalue relation Eq. (A-13) for nonzero y in a future u/2 publication. When y = 0, Y(n(n>> S 16e-Y ± */2 ee-2nir/ U> n=lj2,... (A-16) e,o 21 where y = O5772... is Euler's constant. For u somewhat larger than unity (but 1,1 1 still such that y2 « 1), we have approximately for (j = 0 6^ ~ _—u log _u - JJ, 82 ~ -^ log -r*J - y + -r, so that («) = *i exP [- !%ll ± JL.] exp [-2(n - i.) 1] , n-1. 2,... . It will be seen that the largest eigenvalue is the even eigenvalue Y • e The eigenvalue relation E-5. (A-15) for the even eigenvalues is in agreement with that given in Refs. 9 and 10 when u = 0. This is because the small Y2-approximation employed in Eq. (A-14) consists of taking only the first term in the expansions of S1^ •'(IXJY) and its continuation Eq. (A-ll). v IV. THE EIGENVALUE RELATION FOR y2 » 1 Putting x = sinh C, y = (sech c)1/2 w(^), where z = ix, Eq. (A-l) transforms into iL". + [ - X - Y2 cosh2? + (u2 - I) sech2 ?] w = 0 . For large values of Y2> we expect the solution w(5) to be highly damped away from 5=0, and we have approximately ¥l + [( - X - i. - Y2 + v2) - (Y2 + y2 - I) C2] w = 0 . 2 k 4 Expressing this equation in the canonical form of Weber's equation by putting E, = 21'2 (y2 + V2 -\)UH C, ^ find 22 and I The lowest eigenvalue for even solutions decaying at E, = ± =» is consequently A = — corresponding to the solution w(£) = exp (- -j-C2). Hence we have the approximate eigenvalue relation when y2 » 1, -\ = Y2 + Y - U2 + \ + °(Y l) • (A-18) REFERENCES 1. A. M. M. Todd, M. S. Chance, J. M. Greene, R. C. Grimm, J. L. Johnson, and J. Manickam, "Stability Limitations on High-Beta Tokamaks," Phys. Rev. Lett. _3R, 826 (1977). 2. C. Mercler, "MHD Stability Criteria for Localized Displacements," Plasma Phys. _2_i» 589 (1979). 3. B. Coppi, J. Filreis, and F. Pegoraro, "Analytical Representation and Physics of Ballooning Modes," Annals of Physics 121, 1 (1979). A. R. Y. Dagazian and R. B. 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