Fundamental heating with stellarator wave modes

PHYSICS OF PLASMAS VOLUME 5, NUMBER FEBRUARY 998 Fundamental heating with stellarator wave modes Department of Physics, Auburn University, Alabama 36849 Received 8 September 997; accepted 3 November 997 A perturbation method is developed to find the structure of Alfvén wave modes in a cylindrical waveguide filled with a cold, collisional, uniform plasma with a vacuum layer between the plasma and a conducting wall when the magnetic field is a superposition of a uniform and an inhomogeneous l field created by helical windings. The influence of the helical field on the wave mode structure is treated as a perturbation. It is shown that the m azimuthal component of a modified m fast Alfvén wave is left-hand polarized in the central part of the plasma. This implies a coupling between the m fast right-hand polarized wave and m slow left-hand polarized waves due to the inhomogeneity of the l fields. The coupling efficiency is examined for different plasma parameters. Results demonstrate that efficient coupling between the modes occurs for appropriate plasma parameters in this model, indicating that efficient plasma heating at the fundamental ion cyclotron frequency is possible in stellarators. 998 American Institute of Physics. S7-664X984- I. INTRODUCTION Experiments on the L- Stellarator and GAMMA- have demonstrated that fast Alfvén waves are efficiently absorbed at the fundamental ion cyclotron frequency in stellarators and mirror machines, and lead to central heating. This is due to coupling between fast launched, m) and slow absorbed, m) modes which are coupled through the inhomogeneity of the l magnetic fields. In an analysis of the GAMMA- result, 3 effects of the l quadrupole fields and some longitudinal periodicity were included, but the approximations were severe enough that only the broad outlines of the theory were evident. For stellarators, there are also effects due to the helicity of the l windings. In this more nearly precise treatment, we include helicity of the l windings and consider a cold, uniform plasma in a straight conducting waveguide with a significant vacuum layer between the plasma edge and the wall, thus including both helicity and boundary effects in addition to those effects included in the earlier analysis. Within these restrictions, no further approximations have been introduced beyond the use of first-order perturbation theory. A similar perturbation technique has been used by Girka et al. 4,5 to study the absorption due to the satellite Alfvén resonances in a periodic magnetic field. We ignore the Alfvén resonances in our model by choosing a uniform plasma density and our major interest is in the coupling between waves which can lead to direct heating of ions in the plasma core due to the fundamental ion cyclotron resonance. The influence of the plasma density nonuniformity on the coupling of the modes is beyond the scope of our consideration, but it is not expected to play a significant role in the fundamental resonance heating. The analysis of Paoloni 6 demonstrated that the m fast wave does not experience a cutoff provided the vacuum layer between the plasma edge and a conducting wall is of sufficient thickness. The m and m fast waves each experience a cutoff frequency which could restrict the propagation to frequencies above the cyclotron frequency so that no coupling to the slow Alfvén wave could be realized. For the parameters of most of our examples, this limitation does in fact prohibit any such coupling, so only the m fast wave is considered as the driving wave. We investigate the mode structure of an m fast wave in its lowest radial mode in a cylindrical, uniform, collisional plasma when the nonuniform magnetic field created by an l helical winding is treated as a perturbation. We discover that the m azimuthal component of the modified by the l magnetic field m fast wave is left-hand polarized in the central part of plasma. This effect can be treated as a coupling between the right-handed m fast wave and left-handed m slow waves and can explain the observed efficient absorption of the excited fast wave in stellarator magnetic fields. Section II contains a complete theoretical development of a perturbation method to find a structure of a wave mode in an infinite cylindrical waveguide filled with a uniform, cold plasma with a vacuum layer between the plasma and the conducting wall when the magnetic field in the waveguide is a superposition of a uniform and inhomogeneous field with rotational translational symmetry. Rather detailed derivations are provided because we believe that they can be useful for the analysis of other similar plasma models. In Sec. III we investigate a mode structure of a modified by an l inhomogeneous magnetic field created by two symmetrically wound current-carrying wires m fast Alfvén wave and its coupling to m modes by comparing the Poynting flux through the plasma cross section of the m azimuthal component to the corresponding Poynting flux of the m unperturbed fast wave. We investigate the coupling for several plasma densities and wave frequencies relative to the fundamental ion cyclotron frequency as a function of the helicity of the windings and the current through them. There are two principal results of the analysis, the first of which is 7-664X/98/5()/486/3/$5. 486 998 American Institute of Physics

Phys. Plasmas, Vol. 5, No., February 998 487 that there is a resonance condition for the coupling given by k m k m m, where k m is the wavenumber for the slow Alfvén wave, k is the wavenumber for the fast Alfvén wave, m is the azimuthal mode number for the slow wave, and m is the corresponding wavenumber for the fast wave and indicates the pitch angle of the helical windings such that /L where L is the length of a field period. The second result is the coupling is very strong even for relatively weak inhomogeneities, indicating that the efficiency of fundamental heating should be high. The parameters chosen for the various examples are relevant for the Compact Auburn Torsatron 7 which is a small laboratory torsatron, but the two principal results are independent of the specific parameters used to illustrate various aspects of the coupling. II. PERTURBATION METHOD FOR FINDING THE STRUCTURE OF STELLARATOR WAVE MODES We consider a perfectly conducting infinite straight cylinder of radius b filled with a uniform plasma out to radius a and vacuum from a to b. The magnetic field inside the cylinder is a superposition of a uniform axial magnetic field, B, and a magnetic field created by two conducting wires which are wound helically on the surface of the cylinder. This magnetic field configuration approximately corresponds to an l stellarator field and a wave mode in such a waveguide with such an inhomogeneous magnetic field we call a stellarator wave mode. We will use solutions for the wave mode structure for a waveguide with only a uniform axial magnetic field 8 for the zero order wave fields and consider the influence of a nonuniform l field as a perturbation. We will also use a cold plasma model with collisions. Our approach can be used for finding the mode structure for a waveguide with any additional nonuniform magnetic field with rotational translational symmetry. A. Waveguide modes in a uniform magnetic field A cold plasma in a uniform axial magnetic field may be represented by a dielectric tensor of the form K K K K K K 3, where for a single ion species plasma, K c 3. Then in a uniform magnetic field the waveguide mode fields with a fixed longitudinal wave number k and azimuthal number m in cylindrical coordinates are of the form EE re ik zm t, where in the plasma, the z components of the fields are given by E p z k m k r m k r, 3 B p z m k r m k r, where j / k j, k, k and can be found from boundary conditions at the plasma edge and on the surface of the perfectly conducting waveguide see Appendix A, k and k are roots of the dispersion equation for a cold magnetized plasma, k 3 k k, 3 with k and kk in the uniform plasma. The perpendicular wavenumbers k and k are generally complex numbers with collisions, so the arguments of the Bessel functions are also generally complex. With weak collisions, the longitudinal wavenumber, k, has a small imaginary part compared to the real part. The remaining components are given in Ref. 8. In the vacuum layer, the longitudinal fields are given by E v z B I m T rc K m T r, B v z D I m T rf K m T r, and the transverse components are easily obtained from these. In these expressions, T k /c and I m and K m are Modified Bessel functions of complex argument. The coefficients B, C, D, and F can be expressed in terms of such that B k 3 f m T a m k a m k a], D g m T a m k a m k a, K pii i i i ci pei e i e ce, C B I m T b K m T b, I F m T b D K m T b, i pi ci K i i ci i pe ce i e ce, pi pe K 3 i i i e. We define the related tensor where f m TrI m TrI m Tb/K m TbK m Tr, g m TrI m TrI m Tb/K m TbK m Tr. 5 All expressions throughout the paper are in SI units. An arbitrary dimensional coefficient of proportionality in the 4

488 Phys. Plasmas, Vol. 5, No., February 998 above expressions for electric and magnetic fields in the given waveguide mode has been chosen equal to unity. B. Field equations and perturbation method The magnetic field in the waveguide is a superposition of a uniform axial magnetic field B and a nonuniform magnetic field created by helically wound wires. In our analysis of the helical windings, we will consider two helically wound wires, placed symmetrically with respect to each other, on the surface of the cylinder with equal currents flowing in the same direction. The direction of the currents can be either parallel or antiparallel to the uniform magnetic field. The winding can be either right-handed or left-handed with respect to the uniform field. Our analytical analysis does not refer to the particular type of helical winding, the only requirement being that the magnetic fields created by these windings have a rotational translational symmetry. We will use our particular type of helical winding in Section II C in order to simplify some integrals and in all other places the type of winding is not significant. We take B z, B r, and B to be the components of the magnetic field, at a point (z,r,), due to a helical winding. Letting and be the polar and azimuthal angles of a resultant magnetic field vector in a local spherical coordinate system connected with a point (z,r,), then sin B /B tot, cos (BB z )/B tot, sin B /B, and cos B r /B, where B B r B and B tot (BB z ) B r B /. In the resulting magnetic field, the plasma dielectric tensor in cylindrical coordinates may be represented by T T 4 T 6 T T 4 * T T 5 T 6 * T 5 * T 3, where T K sin cos K 3 K, T K sin sin K 3 K, T 3 K 3 sin K 3 K, T 4 K cos sin sin cos K 3 K, T 5 K cos cos sin K 3 K sin, T 6 K sin cos cos K 3 K sin, where K, K, and K 3 are components of the dielectric tensor for a plasma in a magnetic field whose absolute magnitude is B tot. The components of the tensor T with an asterisk are obtained from the corresponding component without the asterisk by changing K K in the above formulas. For a collisionless plasma, the asterisk is equivalent taking the complex conjugate. For the helical wires wound on the surface of the cylinder, let zt, t, and rb (t-parameter describe the position of one wire, and zt, t, and rb is the position of the other wire. corresponds to a righthanded winding and corresponds to a left-handed winding. In cylindrical coordinates, components of the magnetic field created by such current carrying wires satisfy the symmetry relation, B z,r, z,r,b z,r, zz,r,z, for any z. Taking advantage of this symmetry, we introduce new variables zz, rr, and z. This variable change leads to the replacement recipe, z, z. In these new variables, neither the coefficients nor the dielectric tensor components of T depend upon z. We may therefore assume the wave field components to be of the form f (r, )e i(kzt), so that /z ik. The Maxwell equations then become E z E ike r ib r, ike r E r E z r ib, r r re r E r ib z, B z B ikb r i c T E r T 4 E T 6 E z, ikb r B r B z r r r rb r i c T 4 *E rt E T 5 E z, B r i c T 6 *E rt 5 *E T 3 E z. 6 7 8 9 All functions in the above equations now depend upon only the variables r and. The dielectric tensor T is slightly different from K, so we let the perturbation be represented by K TK, where it is assumed that K is a first-order quantity so that in Eqs. 6 through 8 below, the sums of the tensor terms on the right with first-order fields are second order of smallness and hence neglected. The waveguide mode of Eq. in coordinates (z,r, ) becomes E z (r)e ik zim,..., with k k m. We then write the fields of a mode of a waveguide with a nonuniform magnetic field in coordinates (z,r, ) in the form k k k, E p z (r)e im E z (r, )e ikz, etc. Considering K, k, and E z, etc. as perturbations, then after substituting these fields into Eqs. 7 through and neglecting the second-order terms we obtain E z E ik E r ib r ike e im, ik E r E r E z r ib ike r e im, 3 4

Phys. Plasmas, Vol. 5, No., February 998 489 r r re r E r ib z, 5 ik m B rm B zm i c K E rm K E m r B z B ik B i c K E r K E ikb i c K E r K 4E K 6E z e im, 6 ikb r m,m i c T m, r rb m im r B rm i c K 3E zm i c T 3m, 4 5 ik B r B r B z r i c K E r K E ikb r i r r rb r c K 4*E r K E K 5E z e im, B r i c K 3E z i c K 6*E r K 5*E K 3E z e im. 7 8 The asterisk on the components of K has the same meaning as the asterisk on the components of T. In order to find the mode fields in the waveguide, it is necessary to find the general solution of the system of Eqs. 3 through 8 and match it with the corresponding fields in the vacuum layer. C. Wave fields in the plasma and vacuum layers We now seek a solution of the system of Eqs. 3 through 8 in terms of a Fourier series, e.g., E z n E zn re in. 9 Then, after substituting expressions of the form of Eq. 9 into Eqs. 3 through 8, multiplying each equation by exp(im )/, and integrating over fromto, one obtains im r E zmik m E m ib rm ike m,m, ik m E rm E zm ib m ike r m,m, r re m im r E rmib zm, im r B zmik m B m i c K E rm K E m ikb m,m i c T m, 3 where k m k mk (mm ), and T m re r K e im md E K 4e im md E z K 6e im md, T m re r K 4*e im md E K e im md E z K 5e im md, T 3m re r K 6*e im md E K 5*e im md E z K 3e im md. 6 7 8 All functions in Eqs. through 5 depend on r only and the derivatives are with respect to r. Taking into account the symmetry of the nonuniform magnetic field created by two symmetrical helical current-carrying wires, one can simplify the integrals in Eqs. 6 through 8 see Appendix B. Equations through 5 are similar to the set of field equations for a plasma in a uniform magnetic field, the only difference being the additional terms on the right-hand side. The general solution of Eqs. through 5 is a sum of the general solution of the corresponding uniform system of equations and a partial solution of the nonuniform system of Eqs. through 5. The general solution of the uniform system is of the same type as introduced in Section II A. For the case when mm, the partial solution of Eqs. through 5 contains two terms, one of which corresponds to the right-hand side terms which are proportional to k and is given by an expression of the form E z k(e z ) k, where the notation (E z ) k denotes the derivative with respect to k, upon which the unperturbed solutions depend, such that (E z ) k E z /k. These partial derivatives can be taken assuming that the coefficients,

49 Phys. Plasmas, Vol. 5, No., February 998 B, C, D, and F do not depend on k. The fact that the corresponding partial solution of Eqs. through 5 is of this form can be checked by taking differentials of the wave equations for the unperturbed fields in a uniform magnetic field considering k as an independent variable. The second terms on the right-hand side of Eqs. 3 through 5 involve the terms T m, T m, T 3m. We now find a partial solution of Eqs. through 5 without the right-hand side terms involving k. Using the expressions for E rm and B rm, E rm m r B zmk m B m E m t m, 9 B rm m r E zmk m E m, 3 we eliminate these components so the set of equations may be reduced to E zm imk m r B zm ik m E m i m B m W jm mj k m 3 mj, W,j,m mj k m 3 Y mj, W jm mj, W,j,m Y mj, W 3jm m mj k mj r k mj i mj, mj W 3,j,m m mj k mj Y r k mj iy mj, mj W 4jm mk m rk mj W 4,j,m mk m mj i mj k m k mj, mj rk mj Y mj i mj Y k m k mj, mj ik m t m, B zm imk m r E zm im B r zm i ik m B m i t m i t m, E m i m r B zm r im imk m B r m im t r m, me m E m 3 3 33 with j, and where mj m (k mj r), etc. Once again, an arbitrary dimensional amplitude coefficient of proportionality in the above formulas has been chosen equal to unity, and mj m / k mj, mj m k mj, j,, k m and k m are roots of the dispersion relation of Eq. 3 with kk m, and the functions C lm (r) are C jm r a rc jm r dr, 39 C j,m r rc j,m r dr, 4 where again j, and B m i 3 m r E zm imk m r E m r B m i t 3m, 34 where t jm ( /c )T jm, j,,3, and m k m. Using the variation of constants method, one can find the partial solution of Eqs. 3 through 34 to be 4 Ẽ zm j 4 B zm j 4 Ẽ m j 4 W jm rc jm r, W jm rc jm r, W 3 jm rc jm r, 35 36 37 B m W 4 jm rc jm r, 38 j where the W ijm are the elements of the Wronskian matrix of the system of Eqs. 3 through 34, C m r k m m m k m 3 k m Y m m m t m t m k m k m irk m Y m m k m 3 m t m, t m rt 3m 4 where (C m ) r is obtained from Eq. 4 by interchanging the subscripts m m so that k m k m and m m, and (C 3m ) r and (C 4m ) r may be obtained from (C m ) r and (C m ) r by replacing Y mj by mj and Y mj by mj.in deriving the formula above, we used the fact that the Wronskian for Bessel functions is m xy m x m xy m x/x, and we also used the fact that k m and k m are roots of Eq. 3, along with other tedious calculations. The choice of the

Phys. Plasmas, Vol. 5, No., February 998 49 limits of integration in Eqs. 39 and 4 guarantee that the wave fields calculated from Eqs. 35 through 38 are finite as r. Now the corrections to the zero-order fields in the plasma for the mth azimuthal component for mm are A m E zm k m ke p z k Ẽ zm, 3 4 B zm A m m kb p z k B zm, 43 D. Boundary conditions and mode fields Taking into account the fact that the tangential components of the electric and magnetic fields across the plasma vacuum surface are continuous, and the tangential components of the electric field are zero on the surface of the perfectly conducting waveguide, after tedious calculations we obtain for mm, A m m ke m m, A m m ke m m, 54 55 A m m ia m E m m r k k m ke p k Ẽ m, 44 where ij im (k j ), j,, and m k mj i m k mj a m k mj a k mj m k mj a ia m A m m k B m k k m m rk kb p k B m, 45 where the notation means that a corresponding expression with subscripts changed such that, etc. and m m, etc., multiplied by m is to be added. For mm they are E zm j k m 3 A jm mj mj Ẽ zm, 46 g mt m a T m g m T m a im a T m k mj, 56 mj T m mj k mj 3 m k mj i mj k m m k mj a m k mj a k mj m k mj a f m T m a c T m 3 f m T m a imk m mj at m B zm j A jm mj B zm, 47 T m k mj, 57 E m j m A jm mj r B m j k mj i A jm mj k m k mj mj ia jm k mj Ẽ m, mj 48 mk ma jm mjb m. mj rk mj 49 The corrections to the zero-order fields in the vacuum layer for the mth azimuthal component are E zm B m I m C m K m k m,m E v z k, 5 B zm D m I m F m K m k m,m B v z k, E m k mm rt m B mi m C m K m i T m D m I m F m K m k m,m E v k, 5 5 B m i B c m I m C m K m k mm T m rt D mi m F m K m m k m,m B v k, 53 where the arguments of I m and K m are T m r and T m k m /c. e m E pv k m at E z pv i g m T a T g m T a B z k m E v z b k bt E v b k, bt 3 ak m T bg m T a e m B pv i f m T a c T f m T a E z pv k m at B z i E v z b k c T a K m T bf m T a, where we have introduced a jump condition at the boundary such that E pv E p (a) k E v (a) k, etc. and m Ẽ m a k mm at m Ẽ zma i T m g m T m a g m T m a B zm a, m B m a i c T m f m T m a f m T m a Ẽ zma k mm at m B zm a. pv pv

49 Phys. Plasmas, Vol. 5, No., February 998 In deriving the expressions for e m and e m, we used the Wronskian for the Modified Bessel functions to be 9 I m xk m xi m xk m x/r. The coefficient matrix of Eqs. 54 and 55 is degenerate. In order for the system of Eqs. 54 and 55 to have a solution, it is necessary that the rank of the coefficient matrix be equal to the rank of its augmented matrix. This condition leads to m m k m m k k e m m k e m m k. 58 As long as the mode fields are proportional to an arbitrary constant, one can take any solution of Eqs. 54 and 55 in order to find the mode structure, the only restriction being that the solution has to be of first order. The choice of different solutions of Eqs. 54 and 55 corresponds to the choosing of different coefficients of proportionality and the difference between the corresponding mode structures is of second order. If we examine the simplest solution of system Eqs. 54 and 55, with m, then ke m m A m. 59 m k The coefficients B m, C m, D m, and F m may then be expressed in terms of A m such that where B m D m f m T a A m k 3 m k ake z pv, K m T a Ẽ zm ake v z b k K m T b g m T a T ke b v i A m m k akb z pv B zm a K m T b a K m T, I m T b ke v z b C m B m K m T b k, K m T b I m T b F m D m K m T b T k ik m T b E b v, E v b k m bt E z v b k E v b k. For mm we find A m A m m, A m A m m, 6 6 6 where we have abbreviated ij im (k mj ). The determinant of coefficients for Eqs. 6 and 6 is not zero, so we may solve for the A jm as A m m m, A m m m, so that we may write 63 64 B m f m T m a k m 3 j A jm mj m k mj aẽ zm a, 65 D m g m T m a j A jm m k mj ab zm a, 66 C m B m I m T m b K m T m b, F md m I m T m b K m T m b. 67 Equations 4 through 49, 5 through 53, 58 through 6, and 63 through 67 define the mode structure which in (z,r,) coordinates is given for each component by equations of the form E z z,r, E zre zm re im mm E zm re i[m mzm] eik kz. 68 III. ANALYSIS OF COUPLING EFFICIENCY BETWEEN AN M FAST WAVE AND M MODES The method developed in Section II can be used to find the approximate mode fields in the waveguide with stellarator-like magnetic fields, and the influence on a mode structure of a nonuniform magnetic field can be investigated. The mode structure analysis can give information about excitation and damping efficiency which is important in the analysis of a particular mode which may be useful in plasma heating. For a particular example of this analysis, we consider the changes to the structure of the m fast wave with the lowest radial wave number in an unperturbed uniform magnetic field. In a uniform magnetic field, the m fast wave can propagate at frequencies both below and above ci, it is right-hand polarized RHP in the central part of the plasma, and left-hand polarized LHP in a small region near the plasma edge. This wave can be effectively excited, but its absorption is small. The m slow wave can propagate for ci, it is LHP in the central part of the plasma, and RHP near the plasma edge. This wave experiences the ion cyclotron resonance and its wave energy is effectively absorbed at frequencies close to ci, but it has a relatively small excitation efficiency or its energy is mostly near the plasma edge since it is so strongly guided by the magnetic field.

Phys. Plasmas, Vol. 5, No., February 998 493 Using the technique developed in Section II, we investigated the structure of the mode which is close to the m fast wave when the nonuniform magnetic field is created by two helically wound wires with currents which are described in Section II B. Our calculations were exact and included surface wave effects. Generally, in such an l magnetic field the mode fields are superpositions of different azimuthal components with mm,,4,... see Eq. 68. As long as we investigate the mode structure of the m fast wave which is slightly modified by the magnetic field nonuniformity, the wave fields for the mm azimuthal component will be almost the same as the wave field for the m fast wave in a uniform magnetic field. Under some conditions, the structure of the wave fields for the m azimuthal component is very close to the structure of an m slow wave in a uniform magnetic field, and this m component can be an m slow wave in either the lowest or any of the higher-order radial modes. In such a case, the mode fields with a nonuniform magnetic field are approximately a superposition of the m fast wave and one of the m slow wave modes with a varying radial mode structure. This amounts to a coupling between modes which are independent in the uniform field but no longer independent in the nonuniform field. We define the coupling efficiency as a ratio of the timeaveraged Poynting vector flux through the plasma cross section due to the m azimuthal wave component to the corresponding Poynting vector flux of the unperturbed m fast wave, given by a ReE m B m *)rdr a ReE B *)rdr. 69 We extend this definition of the coupling efficiency for an arbitrary situation when the wave fields in the m azimuthal component are not similar to the fields of a particular m mode in a uniform magnetic field. Because the structure of the fields in the m azimuthal component is different from that in an unperturbed m fast wave, the generalized coupling efficiency is a measure of the changes in the m fast wave mode structure due to the inhomogeneity of the magnetic field. For our example, we consider a waveguide with fixed radius b7 cm and fixed plasma radius a cm. The FIG.. Dispersion curves for the m fast wave, slow m - - -, lowest radial mode;, next higher radial mode Alfvén waves in a collisionless plasma. a n e 7 m 3, b n e 8 m 3. hydrogen plasma density is uniform out to radius a and zero beyond. The uniform magnetic field is in the z direction with field strength B. T, so ci 9.6 MHz. We define b to be a dimensionless helicity of the windings helicity was introduced in Section II B, I/bB is a dimensionless parameter corresponding to the current through the helical windings ( is the ratio of the magnetic field due to a straight wire with current I at the center of the waveguide to the uniform magnetic field strength. We have assumed that in the cold plasma, collisions are dominantly electron neutral and ion neutral. For our estimates of the collision rates, we have assumed the neutral density to be n n 3 8 m 3, T e ev, and T i ev. Dispersion curves are plotted in Figs. a, b for an m fast wave in its lowest radial mode, and m slow waves in the lowest and next lowest radial modes. The plasma in these cases have electron density of n e 7 m 3 in Fig. a and n e 8 m 3 in Fig. b. Figures a and b show the coupling efficiency vs helicity when n e 7 m 3, / ci.5,.3, e / ci., and i / ci.8. Four possible combinations are considered, namely, the direction of propagation of the unperturbed m fast wave is parallel or antiparallel to the uniform magnetic field B with the current through the helical windings either parallel or antiparallel to the direction of B. The coupling efficiency clearly shows resonant behavior, showing maxima where the value of is such that k m k (m m)/b see Section II C coincides with a longitudinal wavenumber of an m mode in the unperturbed magnetic field, or when k f k s where k f is the real FIG.. Coupling efficiency () vs helicity (). a k B, IB, k B, I opposite to B - --,bk opposite to B, IB, k opposite to B, I opposite to B - --.ck opposite to B, I opposite to B, vs, k B, IB - --.n e 7 m 3, / ci.5,.3, e / ci., and i / ci.8.

494 Phys. Plasmas, Vol. 5, No., February 998 FIG. 3. Coupling efficiency vs helicity at lower collision frequency. k B, IB. n e 7 m 3, / ci.5,.3, e / ci., and i / ci.9. FIG. 5. Coupling efficiency () vs helicity () at higher frequency. k B, IB. a n e 7 m 3, b n e 8 m 3. In both cases, / ci.8,.3, e / ci., and i / ci.8. part of k m and k s is the real part of the longitudinal wavenumber for the slow wave which may be positive or negative since the dispersion equation is quadratic in k ). The location of the longitudinal wavenumbers for the m modes with respect to the location of wavenumber k, corresponding to the unperturbed m fast wave, can be seen from Fig. a. The first and second maxima in Figs. a, b with positive correspond to coupling with m slow waves with the lowest and next higher radial wavenumbers with the direction of propagation parallel to the direction of uniform magnetic field, while the first and second maxima with negative correspond to coupling with m slow waves with lowest and next higher radial wavenumbers with the direction of propagation opposite to the direction of the uniform magnetic field. The positions of the maxima are symmetric with respect to such that k b(m m). The fields in this m azimuthal component are left-handed in the central part of the plasma for all presented in this figure and all subsequent figures. The result demonstrates that for the selected weak perturbation (.3), the coupling efficiency may be so large for certain ranges of helicities that the perturbation limits are exceeded. For this reason, the peaks where are not shown on this and the following figures. Figure c is a combination of Figs. a and b, plotted in order to demonstrate the approximate symmetry between different directions of wave propagation, and directions of current in the helical windings. It shows in the same graph vs when k is opposite to B and I is opposite to B and vs when k B and IB. The approximate coincidence of the curves for small and their similar behavior for larger means that vs for the case k B and IB is nearly a mirror image of the case where k is opposite to B and I is opposite to B. A similar symmetry is found when k B, I is opposite to B, and k is opposite to B with IB. In order to demonstrate the influence of collisions on the behavior of the coupling efficiency, we have reduced the collision rates by a factor of two in Fig. 3, relative to those in Fig. a. One may observe a significant increase of the coupling efficiency it is approximately inversely proportional to FIG. 4. Coupling efficiency () vs helicity () at higher density. k B, IB. n e 8 m 3, / ci.5,.3, e / ci., and i / ci.8. FIG. 6. Here log vs log for different helicities. k B, IB. n e 7 m 3, / ci.8, e / ci., and i / ci.8.

Phys. Plasmas, Vol. 5, No., February 998 495 FIG. 7. Radial profiles of a the wave magnetic field components, Re(B z ), Im(B r ), Re(B )---;andbthe Poynting vector for the m fast wave in the waveguide with a uniform magnetic field. n e 7 m 3, / ci.8, e / ci., i / ci.8, a cm, and b7 cm. the square of the collision rates at the maxima due to the decrease of the collision rates. Now the resonances corresponding to different m modes are more sharp. Figure 4 shows the changes to Fig. a due to an increase of the plasma density by one order of magnitude. The coupling efficiency has similar resonant behavior, but the characteristic range of helicities is now wider due to the increased distance between the axial wavenumbers of the fast and slow modes see the corresponding dispersion curves in Figs. a and b. The positions of the first maxima for and correspond to coupling with m slow waves with the lowest radial wavenumbers, while the other maxima are not due to coupling to single modes. They are superpositions of different closely spaced resonances corresponding to modes with higher radial wavenumbers. Figures 5a, b show the changes to Figs. a and 4 due to an increase in frequency to / ci.8. The reasons for the decreases in the coupling efficiency are partly due to the effective increase in collisional effects as the ion cyclotron resonance is approached and to the wider separation of the axial wavenumbers between the fast and slow waves. Figure 6 demonstrates the dependence of the coupling efficiency as a function of current through the windings with fixed helicities compare with Fig. 5a for a density of n e 7 m 3, all remaining parameters remaining the same as in Fig. 5a. As expected, is proportional to for small with some deviation for larger. The results demonstrate that the coupling is efficient even for relatively small perturbations of the uniform magnetic field. FIG. 8. Radial profiles of a the wave magnetic field components, Re(B z ), Im(B r ), Re(B )---;andbthe Poynting vector for the m slow wave in the waveguide with a uniform magnetic field. n e 7 m 3, / ci.8, e / ci., i / ci.8, a cm, and b7 cm. The amplitudes in this figure are arbitrary. FIG. 9. Radial profiles of a the wave magnetic field components, Im(B r ), Re(B r ), Re(B )---,Im(B ) ; and b the Poynting vector for the m azimuthal component of the perturbed m fast wave. n e 7 m 3, / ci.8,.3, e / ci., i / ci.8, a cm, b7 cm, and.95 near the resonant location in Fig. 5a. The amplitudes in these figures are relative to the unit amplitudes of Figs. 7a, b. In order to demonstrate the changes to the mode structure due to the nonuniformity of the magnetic field in the waveguide, we present wave field profiles for modes in a uniformly magnetized plasma as well as for perturbations caused by the nonuniformity of the magnetic field. Figures 7a, b show the profiles of the magnetic field amplitudes actually, their dominant parts, since the imaginary parts are negligible in comparison to the real parts for B z and B, and vice versa for B r ) and the time-averaged Poynting vector for the m fast wave in a uniform magnetic field for plasma parameters n e 7 m 3, / ci.8, e / ci., and i / ci.8. The wave is RHP for almost all radii in the plasma as can be seen from the figure since ib r B RHP, 7 LHP. The discontinuity in the Poynting vector is due to a sharp plasma vacuum boundary in our model, in which E r is discontinuous at the boundary. Figures 8a, b show the corresponding profiles for the m slow wave with the lowest radial wavenumber in the uniform magnetic field for the same parameters. The wave is LHP for almost all radii in the plasma. The amplitudes in this figure are arbitrary. Figures 9a, b show the changes to Figs. 7a, b due to the perturbing field created with.3 and.95. This selected helicity is close to a resonance for the coupling between these fast and slow waves. At this helicity, the m azimuthal component of the modified m fast wave has a structure very similar to the structure of the m slow wave in a uniform field compare Fig. 9 with Fig. 8. The fields in this azimuthal component are left-handed for almost all radii even though the driven m fast wave is righthanded for almost all radii. It should be noted that the amplitudes in Figs. 9a, b are not arbitrary, but are relative to the amplitudes in Figs. 7a, b. From Figs. through 6, it is apparent that the coupling between fast and slow Alfvén waves is substantial since even for small nonuniformities of the magnetic field, there are helicities for which perturbation theory breaks down, suggesting that the perturbed fields dominate the driving field. In

496 Phys. Plasmas, Vol. 5, No., February 998 TABLE I. Relative excitation efficiencies for the various cases in Figs. a, b. i ( i ) Case Figure.8..55.59 k B, IB a, solid.88.5.8..55.9 k B, I opp. B a, dashed.88.8.88.6.8. k opp. B, IB b, solid.55.4.88..8. k opp. B, I opp. B b, dashed.55.47 order to interpret these cases, one must consider the implications of exciting one of these slow wave modes with an antenna through its coupling to fast waves. For any simple antenna, the k spectrum will generally be broad, and only under extraordinary conditions will the spectrum be narrow enough to resolve any of these resonant coupling cases. If the spectrum of the antenna is effectively flat over the range of one of these figures recalling that for coupling between m and m, (k m k )b/, then an integral of (k ) over k which is the same as an integral over except for a constant is the appropriate measure of the relative excitation efficiency between different m modes which are excited through the coupling with m fast waves in a waveguide with fixed parameters. Taking each of the resonances to be Lorentzian in shape which is highly accurate, we then define a relative excitation efficiency for each mode by d, 7 where the range of the integral is understood to include only one resonance at a time while () includes many, so that if one characterizes each resonance to be individually Lorentzian, then A i i i, 7 a i where A i is an amplitude for the ith resonance, i is its location, and a i is a measure of the width which depends on the collision rates. Thus, the relative excitation efficiency of the m mode with radial wavenumber corresponding to the ith resonance is ia i /a i. The values of calculated by Eq. 7 for the plasma and waveguide parameters presented in Fig. are given in Table I. In spite of the fact that the perturbation limit is exceeded for most of the peaks for which is calculated, the ratios between the for different peaks are independent of the perturbation parameter for small enough ), and they are the appropriate measure of the relative excitation efficiency for the different m modes. The numbers in Table I show that for plasma and waveguide parameters presented in Fig., the excitation of the m slow wave with the lowest radial wavenumber is much more effective than the excitation of the same wave with the next higher radial wavenumber for all four different cases. The reduction of the collision rates by a factor of two leads to an increase of the excitation efficiencies by the same factor, e.g., the values of i in Fig. 3 are twice as large as those in Fig. a. IV. CONCLUSIONS From Eqs. 69 and 7 it is apparent that two effectively distinct processes control the coupling strength between the fast and slow modes. The first depends on the breaking of the orthogonality of the fields by the perturbation so that a finite amount of the Poynting flux from the driven m mode is induced in the m component. In the uniform plasma, any cross-coupling between two different modes vanishes due to mode orthogonality, since every azimuthal and radial mode is independent from every other mode. In the perturbed fields, however, every independent mode is no longer pure in the sense that it is effectively a superposition of many of the modes of the unperturbed system. The relative amplitudes of these components, which may be made up of various azimuthal components and radial modes and consist of both slow and fast waves, depends on the magnitude of the perturbation and its azimuthal character (l in our example, and the stronger the distortion of the unperturbed system, the stronger the effective coupling between what was the dominant mode the only mode excited in the unperturbed system and any of the component modes. Thus, an antenna which might be designed to excite only a single radial mode of an m fast wave mode might have a substantial component of what was a pure m slow wave in the unperturbed system. The second process is a resonant process requiring the matching of the values of k for the two modes. Without any periodicity, this would lead to weak coupling, since the dispersion curves in Fig. indicate that the values of k for the slow and fast waves diverge as one approaches the fundamental ion cyclotron frequency, allowing strong coupling only for low frequencies. With periodicity, however, it is always possible to find a resonance, and for a broad enough antenna spectrum, one may be guaranteed of reasonably efficient coupling even close to the ion cyclotron fundamental. It is the combination of these two processes that accounts for the successful central heating observed on the L- stellarator. It was observed that as the frequency increased toward the fundamental cyclotron frequency, the coupling efficiency decreased, although it remained strong. One of the reasons for this decrease is due to the effective increase of the collisional effects as one approaches resonance for the slow waves. In hot plasmas, where collisional effects will play virtually no role, collisionless damping will play a similar role, and while there may be an optimum frequency, as long as there is a fundamental resonance somewhere in the inhomogeneous field, strong damping should be realized, making the process very effective in plasma heating.

Phys. Plasmas, Vol. 5, No., February 998 497 While the model presented here is very restrictive, having ignored toroidal effects completely and having sharp boundary conditions with a uniform plasma density, the two processes noted above will still be operative in real stellarators. A pure tokamak field, which breaks the uniformity of the field, will not lead to similar coupling because the fast and slow waves in a purely toroidal field remain orthogonal, but more importantly, a tokamak lacks both the periodicity of the helical windings of the stellarator which is required to bring the fast and slow waves into resonance and the l character to couple different azimuthal modes. The general picture of the two processes required for significant coupling is believed to be much more general than this example demonstrates. For example, in a planestratified plasma where there is mode conversion between cold plasma waves and warm plasma waves, there is little or no coupling between the modes if the values of k for the cold and warm modes do not match inside the plasma. Away from the coupling region, the fast modes are typically transverse and the energy flux is primarily carried by the Poynting flux while the slow waves are typically longitudinal and the primary energy flux is carried by the kinetic flux, so there is little coupling far from resonance. As the matching point is reached, however, the dominantly transverse wave gains a longitudinal component, and the dominantly longitudinal slow wave gains a transverse component, so that each component carries a combination of Poynting flux and kinetic flux so that they can couple effectively, and in certain cases, the conversion of wave energy from one to the other is total. In these cases, there is simultaneously an exact resonance between the k s of the two modes. When absorption is included, the resonance is broadened, and the conversion efficiency falls below %. The theory presented here, therefore, is the first step in establishing a general theory of linear coupling between waves in any inhomogeneous medium. The parameters for the examples chosen for the numerical studies are very far from fusion parameters, and carrying out a similar analysis with fusion relevant parameters may be difficult, because there will be a large number of overlapping radial modes and the mode numbers will be intrinsically high because of the larger dimensions and shorter Alfvén wavelength. This may make the interpretation of the data difficult, but on the other hand, it may not reduce the overall efficiency unless the radial mode structure were to have low amplitude where the fundamental resonance occurs, which is unlikely. While the toroidal effects are uninvestigated, it will probably require a three dimensional numerical analysis of one field period with a model antenna to estimate coupling and heating efficiency with any reliability. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy under Contracts No. DE-FG-96ER54365 and No. DE-FC-9ER75678. APPENDIX A: BOUNDARY CONDITIONS AND THE DISPERSION RELATION We often refer to an equation of the form of Eq. 3 as the dispersion relation, which in an infinite plasma is appropriate. Here, however, we refer to Eq. 3 as the dispersion equation which yields two values of k for a single value of k z the subscript z has been omitted virtually everywhere, and then the boundary condition equation determines the value of k z, and this expression we call the dispersion relation since it establishes k z (). In order to find the dispersion relation for k, the standard boundary conditions for the fields at the plasma vacuum boundary for a discontinuous dielectric are used, and the perfectly conducting waveguide boundary condition is used at the wall. Assuming the continuity of the tangential components of the electric and magnetic wave fields at the plasma surface and that the tangential components of the electric field vanish at the wall, one may write the dispersion relation for k in the form Uk Uk, A where m k j Uk j m k j, A using the definitions of Eqs. 56 and 57 where k j, j, are roots of Eq. 3 and these in turn are functions of k, and f m and g m are defined by Eqs. 4 and 5 where the prime denotes a derivative with respect to the argument. If k is a root of the transcendental equation A, then the coefficient is given by m k m k. APPENDIX B: PLASMA DIELECTRIC TENSOR INTEGRALS IN COMBINED AXIAL AND l MAGNETIC FIELDS A3 For the magnetic field created by two symmetrical helical conductors, the integrals in Eqs. 6 through 8 are nonzero for mm,,4,.... The magnetic field components satisfy the symmetry relations B z () B z (), B r ()B r (), and B ()B (), so one can simplify the integrals in Eqs. 6 through 8 such that K e i / md 4 K K sin cos K 3 K ]cos m d, K e i / md 4 K K sin sin K 3 K ]cos m d, K 3e i md 4 / sin K 3 K cos m d,

498 Phys. Plasmas, Vol. 5, No., February 998 K 4e i md 4 / K cos K cos m d i / sin sin cos where m (m m). The expressions for those integrals containing K 4*, K 5*, and K 6* may be obtained from those without the asterisk by exchanging K K and K K. K 3 K sin m d, K 5e i / md 4 sin cos sin / K 3 K cos m d i K sin cos sin m d, K 6e i md 4 / K sin sin cos m d / i sin cos cos K 3 K sin m d, V. A. Batyuk, G. S. Voronov, E. F. Gippius, S. E. Grebenshchikov, N. P. Donskaya, K. S. Dyabilin, B. I. Ilyukhin, I. A. Kovan, L. M. Kovrizhnykh, A. I. Meshcheryakov, P. E. Moroz, I. S. Sbitnikova, V. N. Sukhodol skii, and I. S. Shpigel, Sov.. Plasma Phys. 3, 43 987. M. Inutake, M. Ichimura, H. Hojo, Y. Kimura, R. Katsumata, S. Adachi, Y. Nakashima, A. Itakura, A. Mase, and S. Miyoshi, Phys. Rev. Lett. 65, 3397 99. 3 H. Hojo, M. Inutake, M. Ichimura, Y. Kimura, and S. Miyoshi, Phys. Rev. Lett. 66, 866 99. 4 I. A. Girka, V. I. Lapshin, and K. N. Stepanov, Plasma Phys. Rep., 96 994. 5 I. A. Girka, V. I. Lapshin, and K. N. Stepanov, Plasma Phys. Rep. 3, 9 997. 6 F.. Paoloni, Phys. Fluids 8, 64 975. 7 R. F. Gandy, M. A. Henderson,. D. Hanson, S. F. Knowlton, T. A. Schneider, D. G. Swanson, and. R. Cary, Fusion Technol. 8, 8 99. 8 D. G. Swanson, Plasma Waves Academic, Boston, 989, Chap. 5.5. 9 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions National Bureau of Standards, Washington, DC, 964.