Toroidal coupling of ideal magnetohydrodynamic instabilities in tokamak plasmas

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1 Toroidal coupling of ideal agnetohydrodynaic instabilities in tokaak plasas C. C. Hegna, J. W. Connor, R. J. Hastie, and H. R. Wilson Citation: Phys. Plasas 3, 584 (996); doi: 0.063/ View online: View Table of Contents: Published by the Aerican Institute of Physics. Related Articles Head-on-collision of odulated dust acoustic waves in strongly coupled dusty plasa Phys. Plasas 9, (0) Effects of laser energy fluence on the onset and growth of the Rayleigh Taylor instabilities and its influence on the topography of the Fe thin fil grown in pulsed laser deposition facility Phys. Plasas 9, (0) Halo foration and self-pinching of an electron bea undergoing the Weibel instability Phys. Plasas 9, 0306 (0) Energy dynaics in a siulation of LAPD turbulence Phys. Plasas 9, 0307 (0) Free boundary ballooning ode representation Phys. Plasas 9, 0506 (0) Additional inforation on Phys. Plasas Journal Hoepage: Journal Inforation: Top downloads: Inforation for Authors:

2 Toroidal coupling of ideal agnetohydrodynaic instabilities in tokaak plasas C. C. Hegna Departents of Nuclear Engineering and Engineering Physics and Physics, University of Wisconsin, adison, Wisconsin J. W. Connor, R. J. Hastie, and H. R. Wilson UKAEA Governent Division, Fusion (Eurato/UKAEA Fusion Association), Culha, Abingdon, Oxfordshire, OX4 3DB, United Kingdo Received Septeber 995; accepted 6 October 995 A theoretical fraework is developed to describe the ideal agnetohydrodynaic HD stability properties of axisyetric toroidal plasas. The ode structure is described by a set of poloidal haronics in configuration space. The energy functional, W, is then deterined by a set of atrix eleents that are coputed fro the interaction integrals between these haronics. In particular, the foralis ay be used to study the stability of finite-n ballooning odes. Using for illustration the s- equilibriu, salient features of the n stability boundary can be deduced fro an appropriate choice of test function for these haronics. The analysis can be extended to include the toroidal coupling of a free-boundary kink eigenfunction to the finite-n ideal ballooning ode. A unified stability condition is derived that describes the external kink ode, a finite-n ballooning ode, and their interaction. The interaction ter plays a destabilizing role that lowers the instability threshold of the toroidally coupled ode. These odes ay play a role in understanding plasa edge phenoena, L H physics and edge localized odes ELs. 996 Aerican Institute of Physics. S X I. INTRODUCTION The toroidal nature of tokaak confineent devices coplicates the theoretical assessent of their stability properties. In axisyetric systes, the toroidicity of the equilibriu couples all Fourier haronics with the sae toroidal ode nuber. Additionally, shaping effects associated with ellipticity, triangularity, etc., proinent in any of the present day tokaak experients with divertor geoetries, also have the effect of coupling poloidal haronics. As such, a linear eigenode in a toroidal configuration is represented as a su over all the coupled haronics. For plasa equilibriu with low plasa pressure and inverse aspect ratio 0 p/b and odest plasa shaping, the toroidal coupling paraeter can be taken as a sall quantity. With this approxiation, it is possible to ake analytic progress using a perturbation theory. A procedure for coputing the stability properties of resistive agnetohydrodynaic HD instabilities has been developed using this approach. 3 When the low- ordering is acceptable, it was found that for tearing instabilities, the toroidal ode is coprised of a set of cylindrical tearing odes and the stability properties can be deduced fro a natural extension of cylindrical tearing ode theory. For odes with twisting parity, however, the odes are found to be intrinsically toroidal, despite the perturbative nature of the theory, and the toroidal ode is not constructed fro a su of the cylindrical version of the twisting parity solutions. This conclusion is evident fro an analysis of the singular nature of the ode in configuration space, where the toroidal sideband couplings play a crucial role near the rational surface. For ideal HD instabilities, this low- expansion procedure is not appropriate, since instability thresholds for ideal ballooning odes occur when the toroidal coupling paraeter is of order unity. Ideal HD instabilities can be assessed by the use of the ballooning transfor. 4 The traditional ballooning theory involves a perturbation theory for an asyptotically large toroidal ode nuber n. To leading order the linear eigenode condition can be reduced to an ordinary differential equation in the extended ballooning variable. The actual eigenode is then obtained by suing solutions of this equation to yield the variation in poloidal angle on a particular agnetic surface. The radial variation of the eigenode in real space is then obtained by a higherorder expansion in /n. This ethod has proved valuable for stability analysis of tokaak discharges, since the infinite-n ballooning liit is known to be the ost unstable. However, if one is concerned with understanding the behavior of a toroidal ode with a finite toroidal ode nuber or where the plasa edge boundary conditions ay be iportant, a different approach is needed. In this work, we address this proble by working in configuration space. The goal is to obtain an analytic understanding of finite-n ideal ballooning odes and the effect of toroidal coupling on external kink instabilities. uch progress has been ade in understanding the ideal stability properties of shaped tokaak equilibria by using nuerical ethods. 5 7 In this work, we attept to give analytic insights into these nuerical studies. The basic structure of the analytic foralis is outlined in Sec. II. In Sec. II A, the stability properties of toroidally coupled internally resonant haronics are described using a atrix forulation. As discussed in Sec. II B, the atrix eleents are constructed as interaction integrals of different Fourier haronics. We propose a procedure for coputing these atrix eleents for a odel s--type equilibriu as an exaple. In Sec. III A, the 584 Phys. Plasas 3 (), February X/96/3()/584/9/$ Aerican Institute of Physics

3 theory is extended to include the toroidal coupling of internally resonant haronics with another Fourier haronic, whose rational surface lies outside the plasa region. A unified instability condition is derived for such a ballooning/ kink ode. In Sec. III B, we analyze odes whose last rational surface lies just outside the plasa region. Finally, we relate this work to the ideal HD instabilities believed to play a role in low to high L H transitions, and edge localized odes ELs of type I and III. II. INTERNAL ODES For siplicity, we will consider a liiter plasa where a plasa vacuu interface is assued at the surface qq a. The q profile is taken to be onotonically increasing in the plasa region. Thus, we will be interested in plasa odes with poloidal ode nubers with nq a for a given value of the toroidal ode nuber n. These haronics will be referred to as internal odes since their rational surfaces lie within the plasa region. In Sec. III, this analysis will be extended to include couplings to an external ode whose rational surface lies in the vacuu region. A. Theoretical fraework The perturbed agnetic potential in a toroidal plasa is written as a su of Fourier haronics, e in e i 0 rr C, where and are the poloidal and toroidal angles, respectively. The different poloidal haronics are given by a ode aplitude C and a test function that describes the spatial structure of the haronic centered around its rational surface q(r )/n, and is taken to satisfy a noralization condition. The ode nuber 0 is chosen so that 0 nq 0 0, where q 0 is the agnetic axial value of the safety factor. It is assued that there are resonant surfaces in the plasa with toroidal ode nuber n. Note that we do not include haronics whose rational surface is outside the plasa region. For the pressure gradient-driven odes discussed in this section, these haronics do not provide any destabilizing effect and will be ignored. Substituting the Fourier su into the ideal energy functional W, we obtain the foral structure, W C dx K, where f dx K,, g /p dx K p, p. The operators K and K p are functions of the equilibriu, and, unless specified, are arbitrary. These functions are given in Refs. 3 for a sall aspect ratio expansion. For a odel s equilibriu where the paraeters sr(dq/dr)/q and q R 0 (dp/dr)/b are functions of the flux surface label, the integrals f and g /p are specified in Appendix B. The ters f can be identified with the cylindrical potential energy involved with the presence of a particular Fourier haronic, while the ters g /p describe the geoetric coupling of odes with differing poloidal ode nuber, and satisfy the syetry condition g /n g n/. In order to describe the toroidal eigenode, we now need to specify the test function and the aplitude of each haronic. In the next section, we describe a ethod for choosing the aplitudes by iniizing the energy integral. In Sec. II C, the structure of the test function is investigated. B. Finite-n ballooning stability For the purposes of this section, the equilibriu operators K and K p and the set of test functions are assued to be given. Thus, the set of potential energy ters f and g /p can be treated as paraeters, and the W integral is given as a quadratic su of the ode aplitudes C. We now address how to deterine the aplitudes. Since we are interested in finding an energy extreu, it is natural to specify the aplitudes using a iniization schee. In particular, for a given aplitude of the last Fourier haronic, C, the other aplitudes are deterined fro the conditions W/C 0 for to. These conditions are given by f C r C r g r/ p C p g /p 0. 6 ultiplying these conditions by C / and suing yields the relation C f p C C p g /p 4 5 C C dx K, C C g / 0. 7 C C dx K,, i.e., p W C f p C C p g /p, 3 By subtracting this equation fro W, the energy integral can be written as W f C C C g /. 8 The aplitudes C can be deduced fro the conditions given in Eq. 6. These conditions can be written in a atrix for, Phys. Plasas, Vol. 3, No., February 996 Hegna et al. 585

4 where j H ij C j C S i, S i g i/, 0 and the atrix H is defined by H ij f i, for j, H ij g i/j H ji g j/i, for i j. The aplitudes can be deterined fro an inversion of H. Inserting the solutions into Eq. 8 and suing, the energy integral can be written, after soe atrix algebra anipulations, as det H WC C det H W B, where the atrix H is an atrix defined in a siilar fashion as in Eq.. The superscript B is added to indicate that this condition deterines the stability properties of a ballooning ode with poloidal haronics. The condition that this expression iniizes W is that the second derivative atrix, W/g!C i C j, is positive. This is given by det H 0. Therefore the stability threshold condition is given by the zero of the deterinant of H. As approaches infinity, H has an infinite nuber of identical rows. In this liit, the ode aplitudes are equivalent to within a phase factor, and the arginal stability condition is obtained by requiring that the su of the phase factor ties the row eleent of the H su to zero. The arginal stability condition is given by f g /p cospk0, 9 3 where k is the phase factor, which is chosen to iniize W. A siple exaple of how Eqs. 9 can be used to deterine the stability condition and the ode structure is given in Appendix A. C. Structure of the eigenfunction In this section, we suggest a for for the test functions that are to be used to construct the atrix eleents. In order to specify the test functions in configuration space, we rely on guidance fro solving for the analogous proble in ballooning space. As a specific exaple, we consider a odel s- equilibriu, and construct a test function that has the proper singular behavior near the rational surface for this situation. In order to understand the test function in configuration space, it is iportant to appreciate the singularity structure of the toroidal ode near a rational surface. The asyptotic behavior of resistive HD odes near the rational surface is given by x TE x sgnx, 4 for tearing parity, where is the perturbed electrostatic potential that is related to the agnetic potential by ( nq), and by x TW x, for twisting parity, where 4 D I. 5 6 Here D I is the ercier index, 8 which involves the average curvature and easures the available free energy for interchange instability, and represents the discontinuities in the ideal solutions at r. We now consider the case D I 0. For tearing parity odes in toroidal equilibria, the asyptotic behavior of as x 0 is 3 TE 7 x sgnx, which is what would be expected for cylindrical odes. However, the twisting ode structure as x 0 is quite different fro the cylindrical prediction. As shown in Ref. 3, it is given by x/ TW logxc, 8 where C is a constant. Clearly, toroidal twisting parity odes cannot be coputed by coupling cylindrical eigenfunctions, but rather by coupling odes that exhibit the singularity structure given by Eq. 8. The discussion in the previous paragraph describes the singular structure for resistive odes. For ideal perturbations, the associated the ratio of the sall to large solutions of the odes becoes infinite. This is achieved by setting the aplitude of the large solutions to zero while the sall solution aplitudes reain nonzero. Thus, the ideal ballooning ode eigenfunction in configuration space, when the ercier index D is zero, is given by C 0 logxc as the rational surface is approached. As was shown in Ref. 3, an analysis in configuration space deonstrates that the logarithic singularity generated at a particular rational surface is driven by the toroidally coupled sideband solutions. There have been previous derivations of analytical expressions for ideal ballooning stability boundaries. 9,0 In these calculations, a trial function was used in an energy principle integral expressed in ballooning space. The toroidal nature of the eigenode was introduced by having a sideband toroidal coupling ter in the test function. We will use a siilar procedure, but in configuration space. The Euler Lagrange equations obtained fro a variational for of Eq. yield the following equation: C p p C p p p p C p p, 9 where the operators and p can be deduced fro K and K p. All of these operators for an s- equilibriu are described in Appendix B. Using the for infinite n, where all the ode aplitudes C have the sae aplitude, it is possible to show that Eq. 9 is equivalent to the s- ballooning 586 Phys. Plasas, Vol. 3, No., February 996 Hegna et al.

5 ode equation for the Fourier transfored variable, by aking use of the property (x) p (xp), where is the radial distance between rational surfaces. As deonstrated in Ref. 3, the toroidal coupling operators are responsible for the logarithic singularities induced at the rational surface discussed above. Since we are interested in ideal ballooning instability thresholds, it is iportant to account for the singularities due to toroidal coupling in the test function we use for describing the atrix eleents. The following for for the test function to be used is suggested fro the solution of Eq. 9, C C p C p ˆ p p p C p ˆ p p, 0 where /ˆ is an inversion of the cylindrical tearing operator and is a function with twisting parity to be specified. Near its rational surface, this function has logarithic contributions generated by the toroidal coupling ters. A sipler and analytically tractable version of Eq. 0, which contains the essential toroidal coupling haronics of the eigenode, is given by keeping only the first ters of the sus, FIG.. The boundaries for n ideal ballooning stability as deduced fro the equation f g / g / g /3 g /4 0, where the atrix eleents are given in Appendix B. The two curves correspond to the first and second region stability boundaries for an s- equilibriu. WC W K C f p p C C p g /p C C C ˆ q0 C q C I q, 3 C ˆ. Since the test function itself depends on the ode aplitudes, one ust redo the iniization of W in Eq. 6. This has the consequence that there are odifications and ixing of the types of integrals in Eqs. 4 5, leading to the expressions shown in Appendix B, Eqs. B B5. It is worth noting that the procedure outlined above is not rigorous. However, with a reasonable choice of, the s- diagra of ballooning theory can be represented by an analytic expression see Fig.. III. COUPLING TO FREE-BOUNDARY ODES The theory can be extended to free-boundary instabilities by including the effects of a Fourier ode whose resonant surface lies outside the plasa boundary. A. Inclusion of external odes For this case one can write the eigenode, e in e i 0 rr C, where is the external ode aplitude. Using this extended for of, the energy integral takes the for where W K represents the cylindrical potential energy of an external kink and the ters proportional to I q represent the toroidal coupling of the external Fourier aplitude to the internal ode aplitudes. The reaining ters describe the internally resonant odes and their couplings described earlier in Sec. II. The interaction integrals I q are defined in an analogous way to the g /p, in that they are functionals of the plasa equilibriu and the test functions. To choose the Fourier aplitudes C, we follow the sae procedure as given in Sec. II. We take the interaction integrals f, g /p, and I q and the kink potential energy W K as given, so that W is a quadratic function of the ode aplitudes. iniizing W by taking W/C 0 for tofor a fixed external ode aplitude C yields the equations f C r C r g r/r p C p g /p C I 0. 4 These conditions are analogous to those given by Eq. 6, except that there are conditions for the internal aplitudes, and the effect of the external ode aplitude is described by the interaction integrals I. ultiplying each condition by C / and subtracting the su fro W, we express the energy integral in the for WC W K C C I, 5 Phys. Plasas, Vol. 3, No., February 996 Hegna et al. 587

6 which is siilar to the result given by Eq. 8. Inverting the atrix equation given in Eq. 4, and inserting into Eq. 5, leads to the expression WC W K 4 det H cof i H I I i, i 6 where the cofactors of H enter the su. Assuing that the toroidal coupling of the last internal haronic and the external ode doinate the su, Eq. 6 can be written as WC W K I 0 4W B, 7 where the expression W B describes the stability properties of the ballooning ode with internally resonant poloidal haronics, as given in Eq.. The doinant toroidal coupling integral given by I 0 describes the interaction of the last internal resonant haronic and the external kink haronic. An alternative for for W can be derived by fixing the last internal Fourier haronic C instead of C. For this case, the iniization procedure would give W as WC W B I W K Clearly, this expression gives the sae arginal stability condition as that given by Eq. 7. Equations 7 and 8 are analytic expressions for ideal ballooning-kink instabilities. The toroidal coupling integral I 0 plays a destabilizing role. As the stability threshold of either the internal ballooning ode or the external kink ode is approached, the interaction ter becoes ore proinent. It is possible for the ballooning-kink instability to be unstable when the fixed boundary ballooning ode and the cylindrical external kink ode are independently stable. B. Coupled peeling/ballooning instabilities In Secs. II and III A, a rather general procedure has been outlined, which can be used to study the ideal HD stability properties. In this section, we will use the theoretical forulation to investigate a special case. In particular, we are concerned with external odes whose rational surface is just outside the plasa edge. In order to ake analytic progress, soe siplifying assuptions are ade so that the interaction integral and kink potential energy ters of Eq. 7 can be estiated. We calculate the kink-free energy for a siplified odel, in which the edge parallel current is constant in the plasa region and drops to zero at the plasa vacuu boundary. For this case, the kink W can be written as W K a d K a K q arj /B a, 9 dr nq a a where the last ter is evaluated just inside the plasa edge and the first ter describes the logarithic jup of the perturbed agnetic potential across the plasa vacuu interface at ra. The value of the vacuu field contribution to the first ter in W K is stabilizing and depends upon the position of the conducting wall. In the cylindrical approxiation, this ter is given by [(a/b) () ]/ [(a/b) () ], where rb is the location of a conducting wall. For odes where nq a, the vacuu contribution will be saller than the reaining ters in Eq. 9. To copute the contribution fro the first ter in Eq. 9, the behavior of the eigenfunction in the plasa region as it nears the edge ust be deterined. The stability of highly localized ideal kinks, or peeling odes, has been analyzed previously. The theoretical treatent is siilar to the Suyda/ercier analysis of fixed boundary odes. For the condition nq a, the singular nature of the pressure curvature ter causes the eigenfunction to have the shape K rr 0, 30 in the vicinity of the rational surface, where q(r 0 )/n and the ercier index is defined in Eq. 6; the choice of the ercier index is deterined by requiring that the eigenfunction die away far fro the rational surface. Using Eq. 30, the peeling ode W is given by W K sq arj /B O, 3 nq a where the vacuu field contribution enters in higher order. The peeling stability criterion is given by 4 D I q a Rj Bs 0, 3 where the last ter, evaluated at the plasa vacuu interface, is destabilizing, while the agnetic well ter is generally stabilizing for tokaaks. To be consistent with the tokaak ordering used in the reaining part of this section, the ercier ter is given by D I (pr/b )(q )/s. However, it should be noted that Pfirsch Schlüter contributions, which are not included in Eq. 3, provide an additional stabilizing effect. It now reains to calculate the interaction ter I 0.Todo this we follow the procedure outlined in Sec. II C, where we use a odel s--type equilibriu. The expression for I 0 is given by I 0 dx K, dx ˆ dx ˆ O 3, 33 where the first ter describes the interaction integral of the cylindrical eigenfunctions and the iddle two ters describe the effect of the interaction of toroidal sideband contributions to the eigenfunction. The last integral represents integrands of order 3, analogous to the last seven ters in g / of Eq. B. These ters are negligible in the sall quantity nq a as copared with the ters kept. Since we are 588 Phys. Plasas, Vol. 3, No., February 996 Hegna et al.

7 interested in the case where nq a, it is possible to show that the leading-order contributions of Eq. 33 are given by I 0 dx K, a d dr ˆ ra, 34 where the last ter is evaluated at the plasa vacuu interface and the ters higher order in nq a have been ignored. To ake further analytic progress, the shapes of the and need to be specified. It is also assued that the profile paraeters and s are constant in the integration region of interest. We approxiate nqx/ and nq a (x)/, where x is the radial distance away fro the rational surface q/n and is the distance between the two rational surfaces. For, Eq. 30 is used. Assuing that varies slowly near the plasa vacuu interface, I 0 is given by I 0 a s nq a, 35 where all the ters are evaluated at qq a. In the liit D I 0, I 0 reduces to I 0 a slnnq a. 36 Recalling the noralizations used for and in deriving W K and W B, we obtain the desired result, I 0 s se /s lnnq a. 37 Notice that the peeling ode W K is ore singular in the sall quantity nq a than the toroidal interaction ter. Therefore, unless one is rather close to the ideal ballooning ode stability threshold, the ballooning/peeling ode criteria is doinated by the cylindrical contribution, as given by Eq. 3. Edge ballooning and peeling odes are now unified by the expression WW K I 0 4 W B sq arj /B nq a s 4s 4 e /s ln nq a W B O, FIG.. A characteristic plot of edge ideal-hd instability threshold, as given by Eq. 37. The value of the edge current is plotted against the ballooning paraeter for fixed values for s and q and is assued to be large. The dashed line represents the peeling ode stability threshold, as given in Eq. 3. The ideal ballooning threshold is given by / c, which is a vertical line in this plot. The set of solid curves describing the union of the peeling and ballooning odes are paraetrized by the value of nq a. The lowest curve is evaluated at the nq a exp0.353, where the ballooning effect is the ost proinent. The other curves are evaluated at the values nq a 0.05, 0.0, 0.005, and As nq a becoes very sall, the ballooning effect becoes less iportant. 38 where W B describes the internal ballooning instability. The structure of the arginal stability condition is described in Fig.. The axes are labeled by the value of the edge current and the pressure gradient, as easured by the ballooning paraeter, where s,, and q a are all treated as fixed paraeters. The set of curves is paraetrized by the value of nq a. The ballooning effect is ost proinent at the optiized value of nq a exp The curves are labeled by the values nq a exp, 0.05, 0.0, 0.005, 0.00, and the dashed line is the peeling criterion. As entioned previously, for uch saller than the ballooning threshold value c, the peeling criterion deterines stability, while for larger values of, the ode becoes ore like an ideal ballooning ode. IV. DISCUSSION A ethod for studying the ideal HD stability properties of toroidal plasas in configuration space is introduced. This work provides an extension of previous work concerning toroidal resistive HD odes. The toroidal ode is represented as a su of poloidal haronics characterized by a haronic aplitude and a test function, which describes the radial structure of the haronic around its resonant surface. The haronic aplitudes are deterined by the properties of a atrix coprising a set of test function interaction integrals. The zeros of the deterinant of this atrix deterine the arginal stability of the ode. A test function structure is suggested that has the correct singularity structure at each rational surface. This analysis is extended to include the coupling of haronics whose rational surfaces lie within the plasa region to an external, free-boundary haronic, whose rational surface lies in the vacuu region. Thus, the free-boundary ode is able to access external kink free-energy sources. A unified arginal stability condition is derived in Eq. 7, which includes contributions fro the ideal internal ballooning, the external kink perturbation, and their interaction. The interaction ter is destabilizing and ay cause the ballooning/kink ode to be unstable when the internal bal- Phys. Plasas, Vol. 3, No., February 996 Hegna et al. 589

8 looning and external kink ode are independently stable. This analysis is specialized to a case where the external ode s rational surface lies just outside the plasa vacuu interface, i.e., the peeling ode. In this asyptotic liit, the external kink ode s stability is deterined by the peeling ode criterion, Eq. 3. When the ballooning effect is included the ballooning/kink ode stability properties are given by Eq. 38. The stability properties of edge ideal HD instabilities, as given by Eq. 38 are scheatically described in Fig.. At low values of the ballooning stability paraeter, arginal stability is controlled by the peeling ode criterion, which describes the copetition between the stabilizing contribution of the average curvature with the destabilizing contribution of edge parallel plasa current. As the plasa pressure gradient increases, ballooning effects becoe ore iportant and the ode becoes unstable at a value of soewhat below the ideal ballooning ode threshold. In light of these observations, we suggest a odel for L H physics and ELs. Referring to Fig., we notice that at sall values of edge and sufficiently large edge current, the peeling ode criterion is violated. In this regie, a spectru of peeling ode instabilities is present that would produce large anoalous transport rates. This ight be identified as L-ode confineent. As increases or the edge current decreases, the peeling ode stability boundary is approached. When this boundary is crossed, the peeling ode turbulence will disappear and the anoalous transport will be reduced: the L H transition. Although ost of the peeling odes are stabilized, a few odes ay reain that we ight identify as type III ELs. With the confineent iproveent, the value of will increase and stabilize all of the peeling odes. This region can be identified as the EL-free H ode. As increases further, the ballooning effect becoes ore iportant and the onset of type I ELs can be identified with crossing the ideal ballooning stability boundary at large. In this odel, the poor confineent in L odes corresponds to the existence of an edge current that destabilizes the peeling odes. This suggests that an L H transition ay be caused by reducing the edge current. There does see to be soe evidence of this on the COPASS tokaak, which was able to induce the H ode by current rap-down.,3 Additionally, there are indications that edge currents play a role in nuerical studies of the HD stability properties of edge plasas. 7 An additional experiental observation at the L H transition is the change in the edge radial electric field. The odel presented here is only concerned with the linear stability properties of the ideal HD odes, and a description of the nonlinear evolution of the plasa edge is outside the scope of the present work. However, it ay be that the radial electric field change arises as a consequence of the peeling ode suppression. ACKNOWLEDGENTS C.C.H. is grateful to the UKAEA Governent Division, Fusion, Eurato/UKAEA Fusion Association Culha Laboratory for their hospitality and support during the perforance of this work. TABLE I. Stability thresholds as given by the ratio of g/ f for the equilibriu described in Appendix A. As ore haronics are introduced, the stability threshold asyptotes to the ideal n liit g/ f. The haronic aplitudes are evaluated at the threshold value for the given. arginal stability Aplitudes lg/fl, 3 lg/fl.44,.44, 4 lg/fl.36,.68,.68, 5 lg/fl.55,.73,,.73, 6 lg/fl.,.80,.5,.5,.80, 7 lg/fl.08,.85,.4,.6,.4,.85, The work is supported by U.S. Departent of Energy Grant No. DE-FG0-86ER538 and the U.K. Departent of Trade and Industry and Eurato. APPENDIX A: FINITE-n BALLOONING THRESHOLDS As a deonstration of how to use the atrix inforation given in Sec. II B, we consider a siple odel. We assue that the doinant geoetric coupling is given by the toroidal coupling to nearest neighbor rational surfaces. As such, we take g / 0, and g /p 0 for p. To siplify atters, we also take the cylindrical potential energy ters to be identical for each poloidal haronic, f f and g / g. The H atrix has a tridiagonal for, with the diagonal coponents being f and (H ) ii g/, all other atrix eleents zero. This atrix has a nuber of useful properties. For exaple, it is easy to derive the relation det H f det H g 4 det H, A which is true for all values of f and g. For sufficiently large and g/ f saller than unity, the approxiation W B ( f /)( g /f ) odels the behavior of W B. As the arginal stability condition is approached, W B drops quickly to zero after attaining the approxiate value W B f / when g/ f. Clearly, for, the arginal stability condition is f g0, where the ballooning phase is chosen so that g is axially destabilizing. The arginal stability condition and haronic aplitudes for this odel equilibriu can be found siply fro the inversion of the H atrix. Solving for the arginal stability point yields a stability criterion for the atrix eleent g/ f. Table I lists the arginal stability condition and gives the haronic aplitudes for cases with a sall nuber of haronics. As can be seen, the larger the nuber of odes, the easier it is to destabilize the ballooning ode. Table I also shows that the n liit, g/ f, is approached as ore ode aplitudes are added. Besides those listed in Table I, there are other roots to W B. These correspond to higher haronic cobinations with higher arginal stability thresholds. For exaple, another root to 5 is given by f /g, with haronic aplitudes,,0,,, which corresponds to two sets of uncoupled odes. 590 Phys. Plasas, Vol. 3, No., February 996 Hegna et al.

9 APPENDIX B: CONFIGURATION SPACE OPERATORS The operators ipleented in Sec. II C are all easily derived fro expressions defined in Refs. 3 for an s-type equilibriu. They are reproduced here for copleteness: K, dr d P nq / r nq, dr d P r nq, B K, r s nqnq nq nq r d dr nq nq r d dr, B K, nq r nqnq, B3 where P nq rq nq rp q, B4 B 6 nq r nq, B5 and ters involving the Shafranov shift and (r/r) are not included here, but ay be retained in ore general expressions. The operators are defined by r r r r d dr r d dr P nq / r nq d dr r d dr P r nq, s nqnq nqr d nq dr r d dr s nqnq nqr d nq dr r d dr 4r nq nqnq, B6 nq nq, B7 nq nq, B8 B9 nq 4r nqnq. B0 The atrix eleents generated by using the test function of Eq. gives the expressions where f dxk, P r nq P r nq d dr d dr K,, g / dxk, K K, K,, B K, K,, B g / dx K, K, K, K 3,, B3 g /3 dxk, 3 K 3, 3, K, 3 B4 g /4 dx K 3, 4, B5 ˆ ˆ,, ˆ, B6a B6b B6c ˆ, B6d and siilar expressions for, 3, etc. For n, the ideal stability threshold is given by Eq. 3. Analytic expressions for the stability boundary can be derived using ( 3/ x/r s )e x/r s, f(s)/8 3/ (x)/r s e x/r s, f(s)/8 3/ (x )/r s e x/r s, where r s is the location of the rational surface, nqx/, where is the distance between rational surfaces, r s /s, and and s are taken to be constant over the region of integration. The function f (s) is chosen, as f (s)[4s/(6s )] / is not derived analytically. A ore consistent treatent in which and Phys. Plasas, Vol. 3, No., February 996 Hegna et al. 59

10 are calculated fro Eq. B6 will be reported later. The current gradient ter in B4 will not contribute at large n. For siplicity, we also neglect the agnetic well ter. With these approxiations, B B5 can be evaluated with the results f s s f 4 4s s 8, B7 g / e /s 5 3s 3 s s 3 e /s f 3 4s 3 3 4s 3 8s f & 7 6s 3 5 4s 4s 4, g / e /s 3s 3 s s 4 f & 8 3s 3 3 s, B8 B9 g /3 3 e 3/s f 9 8s 3 s 8s f & 3 s 3 5 4s 3 4s 4, g /4 4 e 4/s f 3s 3 4s 8s 3. B0 B In f, the first ter results fro the stabilizing cylindrical energy of the priary ode. The second ter represents the cancellation between the destabilizing sideband energies, as described by the second through fifth integrals of Eq. B and the stabilizing contribution fro. It is possible to see this cancellation by exaining the asyptotic properties of the eigenode equation near the rational surface for sall values of s. 0 The last ter of f becoes ore iportant at larger, and is responsible for the second stability root. The destabilizing g / ter represents toroidal coupling energy and becoes ore proinent at large s, indicating the broadening of the eigenfunction, which allows easier accessibility to pressure gradient-free energy. The solutions to f g / g / g /3 g /4 0 yield stability boundaries in an s- curve shown in Fig.. J. W. Connor, S. C. Cowley, R. J. Hastie, T. C. Hender, A. Hood, and T.. artin, Phys. Fluids 3, J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Fluids B 3, J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Fluids B 3, J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, R. L. Dewar, J. anicka, R. C. Gri, and. S. Chance, Nucl. Fusion, J. anicka, Phys. Fluids B 4, G. T. A. Huysan, C. D. Challis,. Erba, W. Kerner, and V. V. Parail, in The nd European Physical Society Conference, Bourneouth European Physical Society, Petit-Lancy, 995, Paper P C. ercier, Nucl. Fusion, L. E. Zakharov and K. S. Riedel, JETP Lett. 44, O. P. Pogutse and E. I. Yurchenko, Sov. J. Plasa Phys. 5, D. Lortz, Nucl. Fusion 5, S. J. Fielding, P. G. Carolan, A. Colton, D. Gates, J. Hugill, A. W. orris,. Valovic, and the COPASS-D and icrowave Heating Teas, in Ref. 7, Paper Q Valovic, S. J. Fielding, J. Hugill, A. W. orris,. araschek, W. Schneider, F. Ryter, H. Zoh, the COPASS-D and ASDEX Upgrade Teas, in Ref. 7, Paper R Phys. Plasas, Vol. 3, No., February 996 Hegna et al.