PHYSICS OF PLASMAS 1, 0308 005 To-fluid magnetic isl dynamics in slab geometry. II. Isls interacting ith resistive alls or resonant magnetic perturbations Richard Fitzpatrick François L. Waelbroeck Institute for Fusion Studies, Department of Physics, University of Texas at Austin, Austin, Texas 7871 Received 19 July 00; accepted October 00; published online 1 January 005 The dynamics of a propagating magnetic isl interacting ith a resistive all or an externally generated, resonant magnetic perturbation is investigated using to-fluid, drift-magnetohydrodynamical MHD theory in slab geometry. In both cases, the isl equation of motion is found to take exactly the same form as that predicted by single-fluid MHD theory. Three ion polarization terms are found in the Rutherford isl idth evolution equation. The first is the drift-mhd polarization term for an isolated isl, is unaffected by the interaction ith a all or magnetic perturbation. Next, there is the polarization term due to interaction ith a all or magnetic perturbation hich is predicted by single-fluid MHD theory. This term is alays destabilizing. Finally, there is a hybrid of the other to polarization terms. The sign of this term depends on many factors. Hoever, under normal circumstances, it is stabilizing if the noninteracting isl propagates in the ion diamagnetic direction ith respect to the all or magnetic perturbation destabilizing if it propagates in the electron diamagnetic direction. 005 American Institute of Physics. DOI: 10.1063/1.1833391 I. INTRODUCTION Tearing modes are magnetohydrodynamical MHD instabilities hich often limit fusion plasma performance in magnetic confinement devices relying on nested toroidal magnetic flux surfaces. 1 As the name suggests, tearing modes tear reconnect magnetic field lines, in the process converting nested toroidal flux surfaces into helical magnetic isls. Such isls degrade plasma confinement because heat particles are able to travel radially from one side of an isl to another by floing along magnetic field lines, hich is a relatively fast process, instead of having to diffuse across magnetic flux surfaces, hich is a relatively slo process. The interaction of rotating magnetic isls ith resistive alls 3 11 or externally generated, resonant magnetic perturbations 5,7,1 1 has been the subject of a great deal of research in the magnetic fusion community. This paper focuses on the ion polarization corrections to the Rutherford isl idth evolution equation 15 hich arise from the highly sheared ion flo profiles generated around magnetic isls hose propagation velocities are modified by interaction ith either resistive alls or externally generated, magnetic perturbations. According to single-fluid MHD theory, 9,1 such polarization corrections are alays destabilizing. The aim of this paper is to evaluate the ion polarization corrections using to-fluid, drift-mhd theory, hich is far more relevant to present-day magnetic confinement devices than single-fluid theory. This goal is achieved by extending the analysis of the companion paper, 16 hich investigates the dynamics of an isolated magnetic isl in slab geometry using to-fluid, drift-mhd theory. For the sake of simplicity, e shall restrict our investigation to slab geometry. II. REDUCED EQUATIONS A. Basic equations Stard right-hed Cartesian coordinates x, y, z are adopted. Consider a quasineutral plasma ith singly charged ions of mass m i. The ion/electron number density n 0 is assumed to be uniform constant. Suppose that T i = T e, here T i,e is the ion/electron temperature, is uniform constant. Let there be no variation of quantities in the z direction, i.e., /z0. Finally, let all lengths be normalized to some convenient scale length a, all magnetic field strengths to some convenient scale field strength B a, all times to a/v a, here V a =B a / 0 n 0 m i. We can rite B= ẑ+b 0 +b z ẑ P= P 0 B 0 b z +O1, here B is the magnetic field P the total plasma pressure. Here, e are assuming that P 0 B 0 are uniform, P 0 B 0 1, ith b z both O1. 16 Let = P 0 /B 0 be times the plasma calculated ith the guide-field B 0, here =5/3 is the plasma ratio of specific heats. Note that the above ordering scheme does not constrain to be either much less than or much greater than unity. We adopt the reduced, to-dimensional, to-fluid, drift- MHD equations derived in the companion paper, 16 t = d Z, + J J 0 e d 1+ c V z + d /c J, 1 Z t =,Z + c V z + d /c J, + DY + e d U d Y, 1070-66X/005/1/0308/8/$.50 1, 0308-1 005 American Institute of Physics Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308- R. Fitzpatrick F. L. Waelbroeck Phys. Plasmas 1, 0308 005 U t =,U d,z + U,Z + Y, + J, + i U + d Y + e U d Y, 3 V z t =,V z + c Z, + i V z + e V z + d /c J, here D=c +1 c, U=, J=, Y = Z. Here, c =/1+, d =c d i / 1+, Z=bz /c 1+, di =m i /n 0 e 0 1/ /a, A,B= A B ẑ. The guidingcenter velocity is ritten as V= ẑ+ 1+ Vz ẑ. Furthermore, is the uniform plasma resistivity, i,e the uniform ion/electron viscosity, the uniform plasma thermal conductivity, J 0 x minus the inductively maintained, equilibrium plasma current in the z direction. The above equations contain both electron ion diamagnetic effects, including the contribution of the anisotropic ion gyroviscous tensor, but neglect electron inertia. Our equations are reduced in the sense that they do not contain the compressible Alfvén ave. Hoever, they do contain the shear-alfvén ave, the magnetoacoustic ave, the histler ave, the kinetic-alfvén ave. B. Plasma equilibrium The plasma equilibrium satisfies /y0. Suppose that the plasma is bounded by rigid alls at x=±x that the region beyond the alls is a vacuum. The equilibrium magnetic flux is ritten 0 x, here 0 x= 0 x d 0 x/dx =J 0 x. The scale magnetic field strength B a is chosen such that 0 x x / as x 0. The equilibrium value of the field Z takes the form Z 0 x= V 0 * y /d 1 + x, here V 0 *y is the uniform total diamagnetic velocity in the y direction. The equilibrium value of the guidingcenter stream-function is ritten 0 0 x= V EBy x, here 0 V EBy is the uniform equilibrium EB velocity in the y direction. Finally, the equilibrium value of the field V z is simply V 0 z =0. 1 k d3 0 /dx 3 x d 0 1 =0. /dx The solution to the above equation must be asymptotically matched to the full, nonlinear, dissipative, drift-mhd solution in the inner region. III. INTERACTION WITH A RESISTIVE WALL A. Introduction Suppose that the alls bounding the plasma at x=±x are thin resistive, ith time-constant. We can define the perfect-all tearing eigenfunction p x as the continuous even in x solution to Eq. 6 hich satisfies p 0 =1 p ±x =0. Likeise, the no-all tearing eigenfunction n x is the continuous even solution to Eq. 6 hich satisfies p 0=1 p ±=0. In general, both p x, n x have gradient discontinuities at x=0. The quantity p =d p /dx 0+ 0 is the conventional tearing stability index 17 in the presence of a perfectly conducting all i.e.,, hereas n =d n /dx 0+ 0 p is the tearing stability index in the presence of no all i.e., 0. Finally, the all eigenfunction x is defined as the continuous even solution to Eq. 5 hich satisfies 0=0, ±x =1, ±=0. This eigenfunction has additional gradient discontinuities at x=±x. The all stability x index, 0, is defined =d /dx + x. According to stard analysis, 7 the effective tearing stability index, =d ln /dx 0+ 0, in the presence of a resistive all is ritten as 5 = V p + V n V + V, 6 here V is the phase velocity of the tearing mode in the lab frame V = /k. Also, the net y-directed electromagnetic force acting on the inner region takes the form f y = k VV n p V + V, 7 here t= 1 0,t is the reconnected magnetic flux, hich is assumed to have a very eak time dependence. C. Asymptotic matching Consider a tearing perturbation hich is periodic in the y direction ith periodicity length l. According to conventional analysis, the plasma is conveniently split into to regions. 17 The outer region comprises most of the plasma, is governed by the equations of linearized, ideal MHD. On the other h, the inner region is localized in the vicinity of the magnetic resonance x=0 here B y 0 =0. Nonlinear, dissipative, drift-mhd effects all become important in the inner region. In the outer region, e can rite x,y,t= 0 x + 1 x,t expiky, here k=/l 1 0. Linearized ideal MHD yields 1,J 0 + 0,J 1 =0, here J=. It follos that B. Isl geometry In the inner region, e can rite x,,t = x + tcos, 8 here =ky. As is ell-knon, the above expression for describes a constant magnetic isl of full-idth in the x direction W=, here =. The region inside the magnetic separatrix corresponds to, hereas the region outside the separatrix corresponds to. It is convenient to ork in the isl rest frame, in hich /t0. It is helpful to define a flux-surface average operator, Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308-3 To-fluid magnetic isl dynamics in slab geometry. II. Phys. Plasmas 1, 0308 005 fs,, = fs,, d 9 x for, 0 fs,, + f s,, d fs,, 10 = 0 x for. Here, s=sgnx xs,, 0 =0 ith 0 0. The most important property of this operator is that A,0, for any field As,,. C. Ordering scheme For the purpose of our ordering scheme, e require both to be O1 in the vicinity of the isl. This implies that our scale length a is OW our scale field strength B a is O/W, here W are the unnormalized isl idth reconnected flux, respectively. In the inner region, e adopt the folloing ordering of terms appearing in Eqs. 1 : d =d 1, = 0, = 1 s,+ 5 s,,, Z=Z 0 s,+z s,,, V z =V 3 z s,,, J1+ =J s,,. Moreover, = 0, = 0, c =c 0, i,e = 3 i,e, = 3, = 3, D=D 3, d/dt=d 5 /dt. Here, the superscript i indicates a quantity hich is order d i, here it is assumed that d 1. This ordering, hich together ith Eqs. 11 1 is completely self-consistent, implies eak i.e., strongly sub- Alfvénic sub-magnetoacoustic diamagnetic flos, very long i.e., very much longer than the Alfvén time transport evolution time scales. Equations 1 yield d 5 dt cos = 5 d 1 Z, + 3 J 3 e d 1 1+ 3 Vz c + d 1 /c J + Od 6, 11 0=c V z 3 + d 1 /c J, + D 3 Y 0 + e 3 d 1 U 1 d 1 Y 0 + Od, 1 0= M 1 U 1, d 1 L0 U 1, + M 1 Y 0, + J, + 3 i U 1 + d 1 Y 0 + e 3 U 1 d 1 Y 0 + Od 5, 13 0= M 1 V z 3, + c Z, + i 3 V z 3 + e 3 V z 3 + d 1 /c J + Od 5, 1 here Y 0 = Z 0, U 1 = 1, M 1 s,=d 1 /d, L 0 s,=dz 0 /d. Here, e have neglected the superscripts on most zeroth-order quantities, for the sake of clarity. As indicated, some of the terms are Od, since they operate on quantities hich are only important in thin boundary layers of idth Od located on the magnetic separatrix. In the folloing, e shall neglect all superscripts for ease of notation. D. Determination of flo profiles Flux-surface averaging Eqs. 1 13, e obtain U + d i e Y =0 15 i + e Y Y =0, 16 here = d i e 1+ D i + e. 17 Our ordering scheme implies that d 1. No, e can rite /x, provided that the isl is thin i.e., l. It follos that Ms, = d i e Ls, + Fs,, 18 i + e here d d x dl 19 d d d x L =0 d d x d df =0. 0 Note that Ls, Fs, are odd functions of x. We immediately conclude that Ls, Fs, are both zero inside the isl separatrix since it is impossible to have a nonzero, odd flux-surface function in this region. The function Ls, satisfies the additional boundary condition xl V 0 *y /d 1+ as x /. Here, e are assuming that x. Moreover, the function Fs, satisfies the additional boundary condition xf x /x V 0 V as x / 0, here V 0 is the unperturbed isl phase velocity i.e., the phase velocity in the absence of a resistive all or an external magnetic perturbation in the lab frame. It is helpful to define the folloing quantities: ˆ = /, =, X=x/. The solutions to Eqs. 19 0, subject to the above mentioned boundary conditions, are 0 sv *y 1 Ls,ˆ = d 1+X Fs,ˆ = sv0 V ˆ x 1 dˆ X 1 dˆ X, 1 respectively. Of course, both Ls,ˆ Fs,ˆ are zero inside the isl separatrix i.e., ˆ 1. In riting Eq. 1, e have neglected the thin boundary layer idth, hich resolves the apparent discontinuity in Ls,ˆ across the is- Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308- R. Fitzpatrick F. L. Waelbroeck Phys. Plasmas 1, 0308 005 l separatrix. This boundary layer, hich need not be resolved in any of our calculations, is described in the companion paper. 16 Note that the function Ls,ˆ corresponds to a velocity profile hich is localized in the vicinity of the isl, hereas the function Fs,ˆ corresponds to a nonlocalized profile hich extends over the hole plasma. E. Force balance The net electromagnetic force acting on the isl region can be ritten as 1 f y = k J s sin d, 3 here J s is the component of J ith the symmetry of sin. No, it is easily demonstrated that J s sin = 1 k xj s,, so it follos from Eq. 13 that Hence, J s sin = i + e k d dx 5 d F d x3 df d xf. f y = i + e lim x/ x 5 d F d x3 df d xf 5 =s i + e lim x/ x dx d 1 dxf. 6 x dx Finally, Eq. yields f y = i + e V 0 V. 7 x Equating Eqs. 7 7, e obtain the isl force balance equation: i + e V 0 V = k VV n p V + V W/. x 8 J c = 1 X X 1 d + 1 d dt dˆ M M + d L cos, 9 1 here J c is the component of J ith the symmetry of cos. In riting the above expression, e have neglected any boundary layers on the isl separatrix, since these are either unimportant or need not be resolved in our calculations see Ref. 16. No, making use of Eqs. 18, 1,, e can rite Ms,ˆ = s V0 V 0 EBy Lˆ + s V0 V Fˆ x 30 Ms,ˆ + d Lx,ˆ = s V0 V 0 iy Lˆ + s V0 V x Fˆ. 31 0 Here, V EBy =V 0 iy + V 0 ey /1+ is the unperturbed EB velocity i.e., the EB velocity in the absence of an isl, V 0 iy is the unperturbed ion fluid velocity i.e., the ion fluid velocity in the absence of an isl, V 0 ey is the unperturbed electron fluid velocity i.e., the electron fluid velocity in the absence of an isl. Note that V 0 *y =V 0 iy V 0 ey. Furthermore, V 0 = i V 0 iy + e V 0 ey / i + e see Ref. 16 is the unperturbed isl phase velocity i.e., the phase velocity in the absence of a resistive all, V the actual phase velocity. All of these velocities are measured in the lab frame. Finally, both Lˆ Fˆ are zero for ˆ 1, hereas 1 Lˆ = X Fˆ =1 ˆ dˆ X 1 dˆ X 3 33 This equation describes the competition beteen the viscous restoring force left-h side the electromagnetic all drag right-h side acting on the isl, determines the isl phase velocity V as a function of the isl idth W. Note that the above force balance equation is identical to that obtained from single-fluid MHD theory. 7 in the region ˆ 1. No V = J c cos dˆ 1 3 F. Determination of ion polarization correction It follos from Eqs. 11, 13, 1 that see Ref. 1, here V, hich is specified in Eq. 6, is the effective tearing stability index in the presence of the resistive all. Hence, it follos from Eqs. 9 31 3 that Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308-5 To-fluid magnetic isl dynamics in slab geometry. II. Phys. Plasmas 1, 0308 005 I 1 here dw dt I 1 = 1 I = 1 V 0 V 0 = V + I EBy V 0 V 0 iy W/ 3 I 3 = 1 V 0 0 V I EBy + V 0 iy /V 0 V 3 x W/ + I V 0 V x W/, cos dˆ = 0.83, 1 X X dl dˆ = 1.38, 1 dˆ X X dlf dˆ = 0.195, 1 dˆ 35 36 37 38 I X = 1 X df dˆ = 0.69. 39 1 dˆ Equation 35 is the Rutherford isl idth evolution equation 15 for a propagating magnetic isl interacting ith a resistive all. There are three separate ion polarization terms on the right-h side RHS of this equation. The first second term on RHS is the drift-mhd polarization term for an isolated isl see Ref. 16 is unaffected by all braking. This term, hich varies as W 3, is stabilizing provided that the unperturbed isl phase velocity lies beteen the unperturbed local ion fluid velocity the unperturbed local EB velocity, is destabilizing otherise. The third fourth term on RHS is the single-fluid MHD polarization term due to the isl velocity shift induced by all braking see Ref. 9. This term is alays destabilizing, varies as W 1 the square of the all-induced velocity shift. The second third term on RHS is a hybrid of the other to polarization terms. The sign of this term depends on many factors. Hoever, in the limit of small electron viscosity compared to the ion viscosity, hen the unperturbed isl phase velocity lies close to the unperturbed velocity of the ion fluid, 16 the hybrid term is stabilizing provided V 0 *y V 0 0, destabilizing otherise. In other ords, the hybrid term is stabilizing if the noninteracting isl propagates in the ion diamagnetic direction ith respect to the all, destabilizing if it propagates in the electron diamagnetic direction. The hybrid polarization term varies as W is directly proportional to the all-induced isl velocity shift. IV. INTERACTION WITH A RESONANT MAGNETIC PERTURBATION A. Introduction Let the alls bounding the plasma at x=±x no be nonconducting i.e., 0. Suppose that an even in x propagating magnetic perturbation ith the same avelength as the magnetic isl in the plasma is generated by currents floing in field coils located in the vacuum region beyond the alls. The no-all tearing stability index n is defined in Sec. III A. The coil eigenfunction c x is the continuous even solution to Eq. 5 hich satisfies c 0=0 c ±x =1. In general, this eigenfunction has a gradient discontinuity at x=0. It is helpful to define c =d c /dx 0+ 0. According to stard analysis, 7 the effective tearing stability index, =d ln /dx 0+ 0, in the presence of an externally generated, magnetic perturbation is t = n + c c cos t, 0 here t= 1 0,t is the reconnected magnetic flux, hich is assumed to vary sloly in time, c the flux at the alls solely due to currents floing in the external coils. Furthermore, t is the phase of the isl measured ith respect to that of the externally generated perturbation. Let the phase velocity of the externally generated perturbation be V c. It follos that d dt = kvt, 1 here V=V V c, Vt is the instantaneous isl phase velocity. Also, the net y-directed electromagnetic force acting on the isl takes the form f y t = k c c sin t. Note that, unlike the braking force due to a resistive all, this force oscillates in sign as the isl propagates. B. Determination of flo profiles We can reuse the analysis of Sec. III D, except that e must allo for time dependence of the function F to take into account the oscillating nature of the locking force exerted on the isl by the external perturbation. Hence, e rite here Ms,,t = d i e i + e 0 sv * y 1 Ls,ˆ = d 1+ X Ls, + Fs,,t, 3 i + e x F x F 5 t=0. In order to proceed further, e adopt the separable form approach to solve Eq. 5 hich as introduced justified in Ref. 1. In other ords, e try the folloing solution: Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308-6 R. Fitzpatrick F. L. Waelbroeck Phys. Plasmas 1, 0308 005 t Fs,,t = sf 1 kvtdt sin0 + sf kvtdt. 6 cos0 Of course, F 1 F are both zero ithin the isl separatrix. Furthermore, x F 1 F 0, x F 0, t 7 8 as x /. Here, F 0 is a constant. The above boundary conditions imply that the function Fs,,t corresponds to a velocity profile hich is localized in the vicinity of the isl. Matching to the outer region yields F 0 sin0 t kvt dt = V 0 Vt. 9 Fs,ˆ,t = s V0 V 1 exp cos X X Fˆ X s 1 dv kv dt C. Isl equation of motion exp X sin X Fˆ. X 56 Reusing the analysis of Sec. III E, taking into account the time dependence of F, e obtain f y =s i + e lim x/ x x 1 x xf x x 3 F xf d. t 57 Hence, differentiating ith respect to t, e obtain t 1 dv kv dt = F 0 kvtdt cos0 50 According to the boundary conditions 7 8, the first term on the right-h side is identically zero. Transforming the second term on the right-h side, using the fact that the integral is dominated by the region X 1, e get f y = s t0 XF X dx. 58 X d dt 1 dv kvv kv dt= 0 V. 51 Substituting Eq. 6 into Eq. 5, integrating once in using the boundary conditions 7 8, eget sgnv sgnv d dˆ X df 1 dˆ + X F =0, d dˆ X df dˆ X F 1 = F 0. 5 53 Here, = i + e /k V is the localization scale length of the velocity profile corresponding to the function F. Suppose that x. In other ords, suppose that the localization scale length of the velocity profile associated ith F is much larger than the isl idth, but much smaller than the extent of the plasma. In this limit hich corresponds to the eakly localized regime of Ref. 1, Eqs. 5 53 can be solved to give X F 1 = F 0 1 exp X cos X Fˆ, X F = sgnv F 0 exp X sin X Fˆ. 5 55 Here, Fˆ is specified in Eqs. 33. It follos from Eqs. 6, 9, 50 that Finally, Eqs. 50, 51, 56 yield f y = dv dt + k VV V0. 59 Making use of Eq., the isl equation of motion takes the form i + e dv kv dt + i + e kvv V 0 + k W W c sin =0. 60 Here, W c / = c c. The first term on the left-h side represents the inertia of the region of the plasma of idth i + e /k V hich is viscously coupled to the isl, the second term represents the viscous restoring force, the third term represents the locking force due to the external perturbation. Note that the above equation is identical to that obtained from single-fluid MHD theory. 1 The above analysis is valid provided i + e /k Vx. D. Determination of ion polarization correction here Reusing the analysis of Sec. III F, e obtain J c = 1 X X 1 + 1d dt ˆ MM + d L cos, 61 1 Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308-7 To-fluid magnetic isl dynamics in slab geometry. II. Phys. Plasmas 1, 0308 005 0 Ms,ˆ,t = sv0 V EBy sf y t Lˆ Fˆ i + e Ms,ˆ,t + d Lx,ˆ = sv0 V iy 0 Lˆ sf y t i + e Fˆ. 6 63 Here, use has been made of Eqs. 56 59, as ell as the fact that the polarization term integral is dominated by the region XO1. Finally, Eqs. 3, 0, yield I 1 dw dt = n + W c W cos V 0 V 0 + I EBy V 0 V iy W/ 3 0 k V 0 V 0 I EB + V 0 iy / 3 i + e + I k 16 i + e 3 W W c W c sin sin, 6 here I 1, I, I 3, I are specified in Sec. III F. Equation 6 is the Rutherford isl idth evolution equation for a propagating isl interacting ith an externally generated, resonant magnetic perturbation. There are three separate ion polarization terms on the right-h side of this equation. The first third term on RHS is the drift- MHD polarization term for an isolated isl see Ref. 16, is unaffected by the external perturbation. The third fifth term on RHS is the single-fluid MHD polarization term due to the oscillation in isl phase velocity induced by the externally generated perturbation see Ref. 1. This term modulates as the isl propagates, but is alays destabilizing. The second fourth term on RHS is a hybrid of the other to polarization terms. E. Solution of isl equations of motion Let us solve the isl equations of motion, 1 60, in the limit in hich the externally generated magnetic perturbation is sufficiently eak that it does not significantly perturb the isl phase velocity. Let us also assume that is so small that the isl idth W does not vary appreciably ith isl phase. In this limit, e can rite t = kv 0 t + s sinkv 0 t + c coskv 0 t, 65 here s, c 1 V 0 =V 0 V c. Substitution of the above expression into Eqs. 1 60 yields s W W c V 0 66 e sgnv 0 s, here = i + e /kv 0 is the velocity localization scale length. Averaging over isl phase, using Eq. 65, e obtain cos s, 67 sin sgnv 0 s, 68 sin 1. 69 Hence, the average of the Rutherford isl idth evolution equation 6 over isl phase takes the form I 1 dw dt W V 0 V 0 = n + I EBy V 0 V 0 iy W/ 3 s W c V 0 V 0 1+I EB + V 0 iy / 3 V 0 I 3. 70 The first to terms on the right-h side of the above equation are the intrinsic tearing mode drive the drift-mhd polarization term, respectively, are unaffected by the external perturbation. The next three terms ithin the curly braces are the phase-averaged external perturbation drive, hybrid polarization term, single-fluid MHD polarization term, respectively. It can be seen that the external perturbation drive is on average stabilizing, hereas the single-fluid MHD polarization term is destabilizing. 7 The sign of the hybrid term depends on many factors. Hoever, in the limit of small electron viscosity compared to the ion viscosity, hen the unperturbed isl phase velocity lies close to the unperturbed velocity of the ion fluid, 16 the hybrid term is on average stabilizing provided V 0 *y V 0 0, destabilizing otherise. In other ords, the hybrid term is stabilizing if the noninteracting isl propagates in the ion diamagnetic direction ith respect to the external perturbation, destabilizing if it propagates in the electron diamagnetic direction. V. SUMMARY We have investigated the dynamics of a propagating magnetic isl interacting ith a resistive all or an externally generated, resonant magnetic perturbation using tofluid, drift-mhd theory in slab geometry. In both cases, e find that the isl equation of motion takes exactly the same form as that predicted by single-fluid MHD theory see Secs. III E IV C. Hoever, to-fluid effects do give rise to additional ion polarization terms in the Rutherford isl idth evolution equation. In general, e find that there are three separate ion polarization terms in the Rutherford equation see Secs. III F IV D. The first is the drift-mhd polarization term for an isolated isl is completely unaffected by interaction ith a resistive all or an externally generated magnetic perturbation. Next, there is the polarization term due to interaction ith a resistive all or magnetic perturbation hich is predicted by single-fluid MHD theory. This term is alays destabilizing. Finally, there is a hybrid of the other to polarization terms. The sign of this term depends on many fac- Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
0308-8 R. Fitzpatrick F. L. Waelbroeck Phys. Plasmas 1, 0308 005 tors. Hoever, in the limit of small electron viscosity compared to the ion viscosity, hen the noninteracting i.e., in the absence of a resistive all or external magnetic perturbation isl phase velocity lies close to the unperturbed i.e., in the absence of an isl velocity of the ion fluid, 16 the hybrid term is stabilizing if the noninteracting isl propagates in the ion diamagnetic direction ith respect to the all or external perturbation destabilizing if it propagates in the electron diamagnetic direction. ACKNOWLEDGMENT This research as funded by the U.S. Department of Energy under Contract No. DE-FG05-96ER-536. 1 M. N. Rosenbluth, Plasma Phys. Controlled Fusion 1, A991999. Z. Chang J. D. Callen, Nucl. Fusion 30, 191990. 3 J. A. Snipes, D. J. Campbell, P. S. Haynes et al., Nucl. Fusion 8, 1085 1988. T. C. Hender, C. G. Gimblett, D. C. Robinson, Nucl. Fusion 9, 179 1989. 5 H. Zohm, A. Kallenbach, H. Bruhns, G. Fussmann, O. Kluber, Europhys. Lett. 11, 75 1990. 6 M. F. F. Nave J. A. Wesson, Nucl. Fusion 30, 575 1990. 7 R. Fitzpatrick, Nucl. Fusion 33, 109 1993. 8 D. A. Gates T. C. Hender, Nucl. Fusion 36, 73 1996. 9 F. L. Waelbroeck R. Fitzpatrick, Phys. Rev. Lett. 78, 1703 1997. 10 R. Fitzpatrick, S. C. Guo, D. J. Den Hartog, C. C. Hegna, Phys. Plasmas 6, 3878 1999. 11 B. E. Chapman, R. Fitzpatrick, D. Craig, P. Martin, G. Spizza, Phys. Plasmas 11, 156 00. 1 A. W. Morris, T. C. Hender, J. Hugill et al., Phys. Rev. Lett. 6, 15 1990. 13 G. A. Navratil, C. Cates, M. E. Mauel et al., Phys. Plasmas 5, 1855 1998. 1 R. Fitzpatrick F. L. Waelbroeck, Phys. Plasmas 7, 983 000. 15 P. H. Rutherford, Phys. Fluids 16, 1903 1973. 16 R. Fitzpatrick F. L. Waelbroeck, To-fluid magnetic isl dynamics in slab geometry: I-isolated isls Phys. Plasmas submitted. 17 H. P. Furth, J. Killeen, M. N. Rosenbluth, Phys. Fluids 6, 591963. Donloaded 13 Mar 005 to 18.83.63.111. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp