z. -r^a<o ^o^crt^c SCHOOL OF PHYSICS ) THE UNIVERSITY OF SYDNEY i NATURAL RESONANCES ON RATIONAL SURFACES IN TOROIDAL PLASMAS R.C. Cross SUPP-24 January 1984 NATURAL RESONANCES ON RATIONAL SURFACES IN TOROIDAL PLASMAS R.C. Cross Wills Plasma Physics Department, School of Physics, University of Sydney, N.S.W. 2006, Australia Abstract - The continuous frequency spectrum of the torsional Alfven wave in a toroidal plasma is known to contain gaps in the spectrum which are associated with mode coupling on and near rational surfaces. It is shown in this paper that the frequency spectrum is incomplete and that the gape will be occupied by natural torsional wave resonances due to poloidally and radially localised standing waves on and near the rational surfaces. The natural resonances, previously assumed not to exist, may in fact play an important role not only in RF heating schemes but also in determining the stability of toroidal plasmas. Natural resonances of the ion acoustic wave are also discussed and shown to have many features in common with Mirnov oscillations. I. INTRODUCTION It is known that torsional Alfven waves propagate along magnetic field lines in a plasma as if the field lines were under tension and loaded by the mass of the plasma. In a toroidal plasma, there is an important class of surfaces, known as the rational * surfaces, on which magnetic field lines form closed loops after an integer number of transits around the torus. One might expect that closed magnetic field lines, like stretched strings, should support standing torsional Alfven waves. Conversely, open magnetic field lines, such as those on irrational flux surfaces, should not support standing torsional Alfven waves. Analytical investigations of the propagation of torsional Alfven waves in toroida' geometry appear to show that this simple picture is incorrect. KIERAS and TATARONIS (1982) and also D'IPPOLITO and GOEDBLOfcP (1983) have shown that in a toroidal plasma, the frequency spectrum of the torsional Alfven wave is continuous except for gaps in the spectrum associated with mode coupling of different poloidal modes on and near rational surfaces. For typical tokamak parameters, these gaps are quite large and would otherwise be occupied by natural resonances corresponding to standing torsional waves on the closed magnetic field lines of rational surfaces. These results, which show that the natural resonances are specifically missing from the spectrum, appear to be contrary to intuitive expectation. Furthermore, they are inconsistent with recent experimental results (BORG et al., 1984) which show that closed magnetic field lines do support standing torsional Alfven waves and that these waves may propagate independently of the fast or compressional Alfven wave. The continuous spectrum of the torsional Alfven wave is obtained by a conventional normal mode analysis which indicates that, at frequencies below the ion cyclotron frequency, the fast and torsional Alfven wave types are coupled at Alfven resonance layers in an inhomogeneous plasma. The main point of this paper is to show that there is another mode of propagation of the torsional wave in which this wave is decoupled from the fast wave in an inhomogencous plasma. It was recognised by D'IPPOWTO and GOliDBLOHD (1983) that the gap in the spectrum might possibly be filled by a class of "improper" eigenmodes constructed from wave packets, but this possibility was 3. dismissed by the authors since their results seemed to imply that these modes would be damped. However, experimental results (BORG et al., 1984) show that these modes are not strongly damped and that torsional Alfven packets, guided along closed magnetic field lines, will generate standing waves on these field lines. In this paoer, a physical model of torsional Alfven wave propagation is described which is consistent with the experimental results and which indicates that the frequency spectrum obtained by the above authors is incomplete. The complete spectrum includes the natural resonances of a toroidal plasma. The emphasis in this paper is to present a useful physical picture of torsional wave propagation, rather than a rigorous proof of the existence of natural resonances. The above authors have already demonstrated that solutions cannot be found in terms of conventional toroidal eigenmodes. furthermore, the wave packet model indicates that the radial wave functions are localized but otherwise arbitrary and independent of boundary conditions. In any given experimental situation, the wave functions will depend primarily on the structure of the perturbation which excites the wave packet. For example, the wave packet will be highly localized both radially and poloidally to a small cross section of the plasma if it is excited by a highly localized perturbation. The dispersion relation for Alfven waves in a cylindrical diffuse pinch is examined in Section 2 in order to obtain the Poynting vecl r for the torsional wave in a current carrying plasma (Section 1}. In Section A we consider the unique physical properties of torsional wave packets and examine their behaviour in homogeneous (Section 5) and inhomogencous plasmas (Section 6). The effects of toroidal geometry are considered in Sections 7 and 8. The implications for !<!•' heating and plasma staMlily are considered in .Section 9. 4. 2. ALFVEN WAVES IN CYLINDRICAL GEOMETRY In a normal mode analysis of a diffuse linear cylindrical pinch, we may assume perturbations of the form <-(r) citkz-me-ut) (i) where m is the azimuthal mode number, k is the axial wave number and u is the wave frequency. Alternatively we may adopt a local cylindrical coordinate system (r, e , e ) as defined by APPERT et al., i -*• " (1982a and b, 1983a and b), where e|( is the unit vector locally parallel to B and r * e,= e . We consider a uniform density plasma with a uniform axial field Bz and a uniform axial current density J which produces an azimuthal field B„ << B . Neglecting electron inertia, resistivity and plasma pressure, but including the Hall term in Ohm's law, we find that the differential equations derived by APPERT et al. (1982a and b, 1983a and b) can be reduced to the simple form 2 2^1 3 b„ 3b„ 2 V + ^— • k, .2 r 3r V 0 (2) 3r \L r2 where b.. is the component of the wave magnetic field parallel to B and B=eB = 9 B + z B l(o 6A z * -S - Vo • 7Be + kBz ™ m (k x B) = — B = - B - k B. (4) v~ -'v r o r z 0 v ' k 2 = (F2 - G2)/F (5) -> G = - D k ? T u VA (1 - U J „ 2 B p J Be 0 w 2 _ o V. _= a) „D _ o _ o u p 0 0) . A ~ u0 p " a)C l . B ci " m. O Cl 0 1 v. is the Alfven speed and tu . is the ion cyclotron frequency. Equation (5) represents a dispersion relation for Alfven waves in a uniform density, zero shear, current carrying plasma. The effects of toroidal geometry can be approximated in this cylindrical model using periodic boundary conditions with nX = 2TTR or k = n/R where R is the major radius. The safety factor q = r B /(R B J is independent of r in the constant J model since B. a r. Since D = 2/(qR), the representative solutions with D = 1.0 m~ described in this paper are typical of tokamak conditions where q - 2-4. Equation (5) allows for unstable solutions where to has both real and imaginary parts in the non-ideal MUD limit when \l ^=0. In 2 this paper we are concerned only with the stable solutions, :o > 0. If waves are driven by an antenna with fixed, real . , and radial i boundary conditions are imposed to fix k , then equation (5) indicates that there are four solutions for k , representing two torsional and two compressional modes. Each mode type is split by the effects of the current so that the phase velocities * w/k )( (parallel and anti- parallel to B) are slightly different as shown in Fig. 1. These 2 -2 solutions are given for fixed values of kj_ - 0 or 50 m to illustrate the general features of the dispersion relation at low and moderate k, and for typical tokamak conditions. At sufficiently high k , compressional waves do not propagate at frequencies below the ion cyclotron frequency. Low k comprcssional wave solutions allow this wave type to propagate at frequencies down to zero without experiencing I). wave guide cutoff, provided m =f= 0 and provided the plasma is separated from contact with a metal wall by an insulating or vacuum region. This wave propagates as an M11D surface wave, with k_j_-> 0 as u> •+ 0 (APPERT et al., 1983b, COLLINS et al. 1984) and is the wave type which is used in the Alfven wave heating scheme to drive the Alfven resonance at low wave frequencies. The compressipnal and torsional branches with k =0 and k > 0 are seen to touch, but do not cross in Fig. 1(a). This degeneracy is removed at finite k . The phase velocity of the torsional wave parallel to B tends to zero as Q -> 1 due to the ion cyclotron resonance. The phase velocity also tends to zero as ft •> 0 when k is small. For k > 10 m" we recover the usual MIID result that w/k. = v, at low JL ~ HA wave frequencies. The variation of phase velocity, ui/k.. as a function of k is shown in Fig. 2. The phase velocity is essentially independent of k at high k but changes significantly at low ro and low k where the parallel wavelength approaches the pitch length of the helical field.