z. -r^a

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

!

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. From equation (5) we find that k = D when k, = 0 and u) = 0, corresponding to a parallel wavelength

A = 2Tr/k,j = 1/2 where I is the pitch length. In other words, the parallel wavelength of the torsional wave cannot exceed 1/2 at low k . Since solutions with k , = 0 correspond to uniform fields over the plasma cross section, the torsional wave may propagate as a uniform low speed kink at low wave frequencies. At lower k , the torsional wave connects smoothly to the unstable kink modes of a uniform density, zero shear plasma.

One might question the validity of these solutions as 2 2 u> or k -» 0 since the above equations are valid only to 0(Bo /B ).

However, they agree to second order in l*0/B with numerical solutions 7. obtained for arbitrary B./B and for — "onn plasma density and current density profiles (CRAMER, 1984).

3. POYNTING VECTOR FOR THE TORSIONAL WAVE

The Poynting vector S, averaged over a wave period, has the form

= ^- Re[E x b ] 2u l~ o

^ Re[rC-brbH ) •Si(-bibll) o U

e (bb + b.b. )] (6) H r r A. J- ' where b is the complex conjugate of b. The components ofcan bo evaluated from the solution of equation (2) and the auxiliary solutions

b„ = b J (k,r) (7)

i k, mxi: !'u )b« rl- 3r W

m.b r * V dr (9) k.2 L

These solutions arc not exact due to the fact that m varies with r according to equation (4) . However, under the approximation 2 2 that B-. « B_ , the variation of m, with r is negligible. The y Z Am radial component of - S •is zero since its real component is zero, due to the fact that we have assumed an eigenmode solution with a standing wave in the radial direction. In the present context we are interested in that part of which relates to cylindrical wave propagation in either the positive or negative r directions. Except 8.

for the Bessel function dependence, the numerical amplitudes for

the components of < S >with finite il are the same as those for plane wave propagation in x, y, z coordinates, in an infinite plasma,

and are also proportional to the components of the group velocity

vector in slab geometry. We can therefore establish correspondence i with the slab geometry solutions with the substitutions

2

f9 r « ik r kr = k i - -=2- v (io) r '

to show that

S k F2 r r (11) &« k(F- + G-)

fei. _ (Vr)F (12) Sl\ k (I-2 + G2) where = r S + e S + e, S . It is convenient to combine the r x ex. u u S and S components of < S > to define the total transverse component

S, where

si. k^2 (13) S 2 2 '« k|((F + G )

2 2 2 and S, = S +S J- r ex.

Note that when m ~ 0, S = 0, so there is no propagation of

energy in the poloidal (e ) direction. This corresponds to the slab

geometry case where there is no energy propagation in the y direction

if k a 0. The S , S and S, components arc all negligible compared

with SM at high k . From equation (5) we note that the high k_^ 2 2 solutions occur when F -*• 0 in which case G >> F so 7 -> S s ej_ in G~ r ->• and s 3 s rk k 4 u Kk k u ll J- u i.

Both G and k are independent of k at high k so

S /S„ a 1/k,3 and S. /S„ a 1/k,3.

In the ideal MHD limit and with no plasma current, both S ^ ' r and S are zero since there is no b component associated with the e J» it torsional wavo. The effect of finite ft or finite plasma current is to introduce a small b.. component and a small S component which is comparable with S o;ily at small k . Solutions of equation (13) for typical tokamak parameters are shown in Fig. 3. Comparison of these results with those of Fig. 2 indicates that the effect of finite S is to produce a dependence of the phase velocity, oi/k , on k . One can interpret this result to imply that at high k , adjacent torsional Alfven rays propagate with almost no interaction.

At low k , there is interference between adjacent rays, resulting in a change in the phase velocity parallel to B.

The most significant result of this section is that S is almost totally negligible compared with S under typical tokamak conditions where D ~ 1 m and where X ~ 2TIR at fi ~ 0.1. One can therefore construct torsional Alfven rays or wave packets, from a spectrum of high k^componcnts, which are strongly guided along helical field lines. For example, a packet with transverse dimensions ~ 2 cm will be composed of comprnents with X^~ 2 cm,

-7 k, = 2TT/A ~ 400 m" . For such a packet, S /SM - 10 at Q = 0.1 and t) = 1 m . The properties of guided Alfven rays are examined further in the following sections.

4. ALFVLN WAVli I'ACKliTS

A wave packet, as usually defined, denotes a disturbance which is restricted in its tran.svt'rse dimensions and is therefore 10. composed of a continuous spectrum of k components. The dimension of a wave packet in the direction of propagation is normally assumed to be very small, but in this paper the parallel dimension is arbitrary and may be many times longer than the circumference of a toroidal plasma. The only other wave type which has this property is the slow MUD or acoustic wave. The reason for this is that these wavf types are anisotropic and have the unusual property that the phase velocity, w/k , along magnetic field lines is essentially independent of the perpendicular component, k , of the propagation vector. Consequently, a torsional or acoustic wave packet can be regarded as being composed of a continuous spectrum of plane waves, dominated by high k components, but each frequency component has the same k component. To be more precise, a finite length wave

packet will contain a continuous spectrum of a) and kt components, but if the packet is long, both w and k. may be almost monochromatic.

An arbitrary localised disturbance may also have; strong low k . components but, as shown in Section 3, it is the high k components which are most strongly guided along field lines. A torsional or acoustic wave packet, if it is generated continuously by a localized source, would be regarded as a ray. Alternatively, an infinitely long ray is essentially equivalent to an infinite number of finite length wave packets joined end to end. Although this may appear obvious, it cannot be applied to rays of any ^ther wave type since a packet containing a continuous spectrum of k components generally contains a continuous spectrum of k.. components.

The above picture of torsional Alfven wave packet propagation is clear in the case of a uniform density plasma in a uniform magnetic field. In a nonuniform density plasma, or in a nonuniform magnetic field, such a packet cannot be specified by a unique k component since different sections of the packet will propagate at different 11. phase velocity, u>/k , due to the local variation in Alfven speed.

In general, therefore, a monochromatic torsional ray will not only contain a continuous spectrum of k components, but will also contain a continuous spectrum of k components. As described below, such a ray is not significantly refracted by density or field gradients and neither is there any significant mode conversion to the compressional wave, since the Poynting vector for the high k components is directed almost exactly parallel to B. Such a ray can retain its identity for many toroidal circumferences in a tokamak plasma before it disperses radially or before it is significantly attenuated (BORG et al., 1984).

The transverse width of a guided wave packet is arbitrary since it can be composed of an arbitrary spectrum of k components.

Although field lines probably never close exactly on rational surfaces in real plasmas, due to small transverse magnetic field perturbations or fluctuations, standing waves can still exist on field lines which almost close since constructive interference will occur within overlapping sections of finite width wave packets. The model described in this paper is therefore not sensitive to minor perturbations of the rational surfaces. However, the rational surfaces themselves are very sensitive to minor perturbations by helical current filaments which break up the surfaces into magnetic islands. Natural resonances on rational surfaces may therefore be of significance in regard to the stability of toroidal plasmas as described below.

It has been known for many years that Alfven waves are strongly guided by magnetic fields in the magnetosphere (I'LJER and

LEE, 1967; FEJER, 1981), but this aspect of Alfven wave propagation in laboratory plasmas has received very little theoretical or experimental attention. Guided torsional Alfven waves were first 12. observed experimentally in a laboratory plasma in 1983, using a small antenna to generate a localized Alfven disturbance (CROSS,

1984a). The torsional Alfven wave packet launched by the antenna was observed to propagate along field lines as a radially localized disturbance for distances up to about 20 cm from the antenna, but the packet diffused radially across field lines as it progressed in the axial direction, as a result of resistive diffusion. The high k components of the packet were more heavily damped than the low to components. The plasma temperature in this experiment was only

1.3 eV. Typical results from this experiment are shown in Fig. 4, including for comparison results demonstrating isotropic propagation of the fast wave and guided propagation for the torsional and acoustic waves. The acoustic wave results are described further in

Section 9.

In high temperature toroidal plasmas, torsional wave packets should propagate many times toroidally before they disperse radially as a result of resistive diffusion. Indeed, such packets have recently been observed, in a low temperature (~ 10 eV) tokamak afterglow plasma, to propagate for distances beyond 300 cm without significant dispersion or mode conversion to other wave types

(BORG et al., 1984). In this experiment, a small dipole antenna was inserted in the plasma and driven continuously at frequencies

0.1 < oj/u) . < 1. The dipole was aligned to preferentially excite a torsional wave, but there was some evidence in tKe small baseline signal, that a fast wave mode was also excited. Standing torsional rays were observed, in the afterglow plasma, with transverse dimensions ~ 20 mm, and with a transverse structure similar to that of the vacuum field of the dipole. In other words, the ray had the characteristics of a highly directional laser beam but with a transV'. se mode structure modified from the ideal vacuum dipole field 13. by the effects of radially localized wave interference. These results have established conclusively that the simple intuitive picture of torsional wave propagation is essentially correct.

In the following sections, we examine the propagation of torsional rays in cylindrical and in toroidal plasmas.

5. LOW FREQUENCY BEHAVIOUR OF TORSIONAL WAVES

The concept of ray propagation is commonly applied to high frequency waves in plasmas but has not previously been applied to torsional Alfven waves in current carrying plasmas. To illustrate the usefulness of this concept, we consider in this section the propagation of low frequency torsional cylindrical eigenmodes in a uniform density plasma with zero shear. Such modes are of little practical interest, because density and shear profiles are generally not uniform. Furthermore, we ignore periodic boundary conditions in this section. However, the modes considered here are of theoretical interest since they highlight the physical properties of Alfven rays.

The boundary conditions for a cylindrical torsional wave eigenmode in a uniform density plasma are satisfied if k a ~ 3.8 where a is the plasma radius, so k > 20 m for most experimental conditions. This result applies whether we assume either metal wall or vacuum boundary conditions. As shown in Fig. 2, the dispersion relation for such an eigenmode is, to a good approximation, given by a) = t k v., the sign depending on whether k, > 0 or k < 0.

Hence from equation (3)

/ B mB £0 _ « / if k > 0 (14) k _v° - __or . 7 A w H 14.

mB - B if k 115) or r.j

where co/k is the phase velocity in the z direction and is independent

of r since BQ a r. The torsional wave therefore has an isolated low

frequency cutoff, where k = 0, and a preferred direction of propagation at frequencies below the cutoff frequency. Strictly,

there is no cutoff since there is wave propagation either side of the "cutoff". These features, consistent with the results reported by APPERT et al. (1983b), appear to be most uncharacteristic of the torsional wave. There is a "cutoff" at w = w. = ± m v, B /(rB ) depending on the sign of mB,. If mB , > 0 then a low frequency (w < :,j ) torsional eigenmode can propagate only anti-parallel to B , but if

w " (jj the mode can propagate in either direction. If mBfl < 0 then a mode with OJ < u can propagate only parallel to B while w > to modes can propagate in either direction. At u = u) the wave fronts are parallel to B and rotate purely in the 0 direction, but with a phase velocity w/k = v. along helical field lines.

The significance of the sign of mBft when field lines are helical can be understood as follows. If we reverse the direction of current in a plasma we will reverse the sense of the helical twist in the field lines. Consider the case where the conventional direction of plasma current is parallel to B. An observer viewing along B sees the field lines twist clockwise as on a right hand thread. As shown above, a torsional wave eigenmode can be regarded as being composed of adjacent, magnetically guided, non-interacting rays if k > 20 m" .

We now show that the plane of polarisation within each ray will rotate

in space and time, as it travels along helical field lines, to follow

the twist of the helical field.

Consider, for example, an eigenmode with k > 0 and with 15. magnetic field perturbations of the form b = f (r)cos(kz+m'J-ut). If m > 0, the global field pattern rotates clockwise in time at any given z but anticlockwise with increasing z at any given time.

A maximum in b occurs at 6 = (tot-kz)/m. The plane of polarisation therefore rotates by an angle A0 = (wt - kz )/m in a distance z. after a time t . During this time, a k > 0 ray advances a distance v.t, along a helical field line or a distance z, = v.t,B /B in the A 1 lAlzo

z direction. From equation (14) we find that A8 = z..Bfi/(rB ), which coincides with the twist in angle of a helical field line in a distance z..

The unusual behaviour of low frequency torsional wave eigen- modes can be attributed to the propagation of guided rays whose plane of polarisation rotates to follow the twist in the helical field.

Since this behaviour is independent of wave frequency, but the rate of rotation of the plane of polarisation varies with u) at fixed z, the resulting wave fronts may propagate cither with k > 0 or with k < 0, depending on as. When mB > 0 and u> < u., the torsional wave travels as a backward wave, with individual rays carrying energy in the positive z direction, but with w/k <• 0.

In the following section we consider the effects of density gradients and shear. Except for a narrow range of frequencies near the lower edge of the Alfven continuum, global torsional cigcnmodcs do not exist because the wavefronts are distorted by local variations in ray phase velocity and the plane of polarization is distorted by shear.

b. HI'HECTS OH DENSITY GRADIENTS AND SHEAR

It is commonly but incorrectly assumed (STIX and SWANSON, 1983) that torsional waves are refracted by density gradients. To demonstrate that this is incorrect, consider a plane parallel wave incident on the 10. boundary between two uniform regions of plasma, one of density p.

and one of density p9. Suppose that the density jump is transverse to the direction of a steady field B which is uniform and equal on both sides of the boundary. If 0, is the angle between the incident

k. and B, and 0- is the angle between the refracted k9 and B then a simple application of the usual boundary conditions yields Snell's law

v v 1 2

cos B1 cos 82

where v7 = ;o/k and v., = u/k-,. In the MUD limit and for zero plasma current, v = v. for the fast Alfven wave, v = v. cos 0 for the torsional wave and v = a cos 9 for the slow MUD wave where a is s s the acoustic speed. The expression for the fast wave is valid in a low 3 plasma where v. >'•• a . We find the surprising result that :>, = As 1 for the acoustic and torsional wave types. In other words, there ii" no refracted wave since the group velocity vector for these wave types allows energy to propagate only parallel to B. The same result is obtained if we assume a jump in the B field. The effects of finite frequency or finite plasma current are to produce a small transverse component of the group velocity vector, in which case there is a refracted beam solution if the density jump is small, but the refracted beam carries almost no energy for high k incident and refracted waves.

The qualitative features of fast Alfven wave propagation, and some of the features of torsional wave propagation, in a nonunifo density, finite shear, cylindrical plasma, can be understood in the

WKB approximation by plotting k vs r for assumed density and q profiles for a fast wave cigpnmodc of fixed m and n. Such plots are very sensitive to the assumed profiles and to the signs of m and n, but the essential features can be illustrated by the relatively

simple schematic example shown in Fig. 5. This example is for

typical tokamak conditions where n = +1, m = +1, SI - 0.1 and for parabolic density and current density profiles. The Alfven resonant 2 surface is located where k = <*>, or from equation (5), where

u .2 z m (16) 2 2 = k = —=—=2 2 • n +q v (v l - Si ) •» R B <• J AA ' o

Driven torsional waves propagate in the narrow region between k = 0 and k , = <=° on the low density side of the resonance and the

fast wave propagates as a low k surface wave, if m^ 0, on the high density side (APPERT et al., 1983b; COLLINS et al., 19S4). In general, and regardless of the signs of m and n, the region between k = 0 and k = °° which supports torsional waves is typically only a few cm wide.

In the MUD limit, and for zero plasma current, this region is infinitely thin. With finite plasma current or at finite frequency, this region has finite width and represents a zone where adjacent torsional rays, propagating at the local Alfven speed -/k.. parallel to B, huve the same phase velocity co/k in the z direction, as assumed for the eigenmode of given m and n.

The right hand side of equation (16) is independent of density and, for a given m,n mode, depends on the q profile. The

left hand side depends on the density profile. If the profiles are adjusted so that the left side remains slightly less than the right

1 side over most of the plasma cross section, then the resonance may be absent from the plasma but k will be large over most of the plasma cross section. Under these conditions, which can occur over a narrow

frequency range near the lower edge of the Alfven continuum, the torsional wave will propagate over a wide cross section of the plasma id.

as a global eigenraode (APPERT et al., 1982a, 1983b). However, near the edge of the plasma where v. -* ">, the wave fields are evanescent under most conditions of interest.

Within the zone of propagation of torsional waves, the flow of energy is almost totally parallel to B, even in the presence of strong density gradients. In order to couple energy into the

Alfven resonance by means of an external antenna it is necessary to drive the resonance with a fast wave, linergy is fed into the torsional wave by mode conversion of the evanescent wave on the low density side of the resonance layer or mode conversion of the fast wave via the evanescent zone on the high density side of the resonant layer. Details of the mode conversion process have been described by STIX and SWANSON (1983) and by WINGLEE (1982). This is a practical scheme for coupling energy into a resonance layer near the centre of the plasma, but it is not the only method of coupling energy to torsional Alfven waves. Consequently, an analysis of t' e fast wave driving scheme yields only part of the total spectrum of torsional Alfven wave behaviour. The complete spectrum is described below.

7. EFFECTS OF TOROIDAL GEOMETRY

It is well known that fast waves propagate isotropically in toroidal geometry in essentially the same manner as in cylindrical geometry and that cavity eigenmodes can be excited by an external antenna. These waves can therefore be used to couple to an Alfven resonant layer in a torus by means of an external antenna. One might expect that the strongest forced resonances in toroidal geometry would be those corresponding to the natural resonances on the rational surfaces. However, the results of the toroidal calculations show that these resonances are completely decoupled from the fast 19.

wave. Indeed, the general conclusion of D'IPPOLITO and GOfcDBLOLD

(1983) is that these resonances cannot exist. In the periodic

cylinder model, the Alfven resonance condition is defined by

equation (16) . Modes defined by (ra,n) and(m',n) are degenerate

if (m + nq) = - (m1 + nq). For example, on a q = 1.5 surface,

the (-1, 1) and (-2, 1) modes are degenerate. As shown by KIERAS

and TATARONIS (1982), the effect of toroidal geometry is to couple

these modes on and close to rational surfaces at frequencies which

are significantly different to those given by equation (16), thereby

creating a gap in the continuous spectrum of k = °° solutions.

The degenerate modes have equal and opposite k values and

are therefore degenerate under circumstances where guided Alfven

rays, travelling in opposite directions on closed field lines with 2 1 phase velocity o/k,. = ±(1 - f: ) v., can interfere to form standing waves. Tor example, on the (| = 1.5 surface, k .. = + B^/3RB for the

(-1, 1) and (-2, 1) modes, corresponding to a standing wave with

X ., = 6TTRB /B^ in the cylindrical model or with a parallel wavelength equal to the total length of a closed field line on the q = 1.5

surface of a torus.

The gap in the frequency spectrum has been explained by

D'IPPOLITO and I'.OllDBLOIiD (1983) and by KIliUAS and TATARONIS (1982) by analogy with the gaps which appear in the energy spectrum of a single electron in a periodic potential lattice. In the case of a torus, the magnetic field varies periodically along closed field lines.

It is well known (BRILL0U1N, 1946) that in any periodic structure, wave propagation is possible only in certain pass bands and that these bands arc separated by stop bands in which wave propagation

is evanescent. The upper and lower frequencies of a given stop band are determined by the condition that standing waves are formed with 20.

an integer number of wavelengths in the period, d, of the structure.

An exception is the fundamental, or lowest frequency stop band, for which the upper and lower frequencies corresponding to standing waves with a fundamental wavelength component. A, equal to 2d.

One dimensional wave propagation in a periodic structure can be modelled by the Mathieu equation (BRILLOUIN, 1946, McLACHLAN,

1964)

72 * 1 y = 72tJ [Lo ' 2S cos TTj y - ° "'> 9x v 3x *• ; where v is the local phase velocity, w is the wave frequency and C , C. are constants which define the periodic variation in v. This equation can be used to model the formation of standing waves which are localized both radially r.r.d pcloidally along adjacent closed helical field lines as a result of guided ray propagation. In a torus, the toroidal field varies periodically along closed field lines between a minimum value B, = B R /(R + r) and a maximum value 1 o o o B. = B R /(R - r) where R is the major radius and r is the minor 2 o o o v radius of the rational flux surface under consideration. For small r/R , this situation is modelled with Cj/C = r/R in equation (17) and with x equal to the distance along a field line measured from a point where the toroidal field has the value B_.

Consider now a q = 1,5 surface where a field line makes three toroidal and two poloidal transits in a closed loop. The fundamental resonance frequencies, with one wavelength in the length of a closed field line, corresponds to a case where A = 2d and where d ~ 3TTR is the periodic length of the variation in the toroidal field on a closed field line. The relevant solutions of the Mathicu equation are n = l ± 0.5 Y - f2- • 0(.,°J

2 2 i J 2 2 whe- ^ n = u> iiC /TT" and 7 = 2ui~d C./ir and where y/n = 2Cj/C = 2r/R < 0.4 for typical tokamak conditions. The fundamental resonance frequencies for a natural 2 resonance on a q = 1.5 surface are therefore, to 0(Y ),

o 1 • -£- (18) ~ 2Ro

2 2 2 where too = tr /(d Co ) . The wave field variation at the lower frequency is given by the infinite scries solution

E TTX Y 3TTX ,,, 2~I in x -16sin ~r *0(Y 3J where y is an arbitrary amplitude constant and where n - 1 and 2 Y - 2r/R when (r/R )* << 1. The wave field dependence on x at the higher frequency is given by

7rx r Y cos 3^ x+ ... 2 ~ y = yx [cos T " T7J ~ °(V J_ where y. is an arbitrary amplitude constant. These solutions show that the wave fields, at the lower frequency, are a maximum at x = 0.5 d and x = 1.5 d where the toroidal field is a minimum and that the nodes are located at x = 0 and x = d where the toroidal field is a maximum.

The higher frequency solution is orthogonal with field maxima at x = 0 and x = d and with nodes at x = 0.5d and x = 1.5d. In general, the helical pitch of a given field line varies with poloidal angle, but it is easy to demonstrate that the nodes and antinodes are equally

paced in toroidal angle, $, around the toroidal circumference, with spacing A<{> = rr/2. liquation (18) has been derived tor a q = 1.5 surface but

is independent o^" q and applies equally well to a natural resonance

on any rational surface when the fundamental wavelength component X

is equal to 2d. At higher frequencies or on other rational surfaces

such as the q = 2 or 3 surfaces where the fundamental wavelength is

equal to d, the solutions of the Mathieu equation yield twe resonant

frequencies

5r id = 2(i) I and w = 2u 1 + (19) 6R 6R o )

2 2 when r « R . Solutions (18) and (19) are specific to the functional

variation of phase velocity assumed in equation (17) and, as such, do

not model precisely the variation of Alfven speed on closed field

lines in a tokamak plasma. Nevertheless, the model is a good approximation, especially since the solutions (18) and (19) depend primarily on the amplitude of the velocity perturbation and are not particularly sensitive to the precise functional variation of phase velocity. For example, the frequency splitting given in (18) is

increased to

Mathieu equation (BRILLOUIN, 1946).

A significant feature of equation (18) is that the frequency

splitting is relatively small for typical tokamak conditions where r/R < 0.2, and is about ten times smaller than the splitting found by Kil-RAS and TATARONIS (1982) for Alfven resonant surfaces coincident with a q = 1.5 rational surface. These authors obtained solutions of

the form <>) = '.) (1 ± eA) where c = a/RQ and A = 2r(2 + A)/a, valid in the large aspect ratio limit z « I. Mere, A = 3„ + (W2) - 1 is the asymmetry factor describing the variation of poloidal field with poloidal angle on a flux surface and is typically about 0.5 when

the poloidal beta 3H ~ 1 and where 1./2 = 0.5 for a parabolic

plasma current distribution. Consequently, the frequency splitting,

representing the size of the gap in the continuous spectrum on a

q - 1.5 surface is

(20)

Although this solution is valid only in the limit r << R , the

indication is that the gap in the continuous spectrum for realistic

tokamak conditions is relatively large and will be occupied by natural

torsional wave resonances with a much smaller frequency gap.

The difference between the results obtained in this paper, as expressed in equation (18), and the result obtained by KIERAS and

TATARONIS (1982), expressed in equation (20), is due to the fundamental difference between an Alfven resonance and a natural resonance. A natural resonance can be driven only in the immediate vicinity of a rational surface. An Alfven resonant surface can be driven on any flux surface in a toroidal plasma, including a rational surface, but not at the natural resonance frequencies. Both types of resonance, when driven on a rational surface, can be described in terms of standing torsional Alfven waves along closed field lines with a fundamental wavelength component equal to the length of the closed field lines. Despite these similarities, the differences are readily apparent.

An Alfven resonance driven by a fast wave in a toroidal plasma can be described analytically as a toroidal eigenmode, characterised by an integer toroidal mode number, n. Such solutions cannot be found in toroidal geometry for a pure poloidal mode number, m. The effect of toroidal geometry is to couple degenerate poloidal modes, as shown 2-1. analytically by KIURAS and TATARONIS (1982). Similar results have been obtained numerically by APPERT et al. (1982c) using a toroidal code to examine Alfven resonant surfaces coincident with irrational flux surfaces. These surfaces are driven by mode conversion of the fast wave at all points within the surfaces. The energy flow is directed primarily along field lines, so that these surfaces coincide with flux surfaces, not only in cylindrical geometry but also in toroidal geometry. This last result was proved rigorously by PAO

(1975) in the MHD approximation. The results of this paper indicate that Alfven resonant surfaces will also coincide with flux surfaces at frequencies approaching the ion cyclotron frequency.

A natural torsional wave resonance occurs as a result of guided propagation of finite k , torsional rays which can propagate independently of the fast wave and without mode conversion to the fast wave. Mode conversion of one wave type to another requires the two wave types to locally have the same wavenumber at the same frequency. The torsional and fast Alfven wave types arc coupled by mode conversion at all points within Alfven resonance layers. On a q = 1.5 surface, for example, mode conversion occurs at either of the two well separated frequencies given by (20). Since these frequencies differ from those given by (18), but the wavenumbers are the same, mode conversion will not occur on the q = 1.5 surface at the natural resonance frequencies. The results obtained by KIERAS and

TATARONIS (1982) for the gaps in the continuous spectrum pre specific to those degenerate modes having the same toroidal mode numbers and with poloidal mode numbers, m, differing by one such that m + nq = t 1/2.

The results are applicable to q = 1.25, 1.5, 2.5 surfaces for example, but not to q - 2 or 3 surfaces. Nevertheless, one can argue on physical grounds that mode conversion cannot occur on any of the rational surfaces at a natural resonance frequency. 25.

As described in Section 9 below, guided rays can form

standing waves on adjacent closed field lines, with transverse

dimensions typically only one or two cm wide in a tokamak plasma.

The wave fields are both radially and poloidally localized within

any given plasma cross section, near those points centred on a given

helical field line intersecting the cross section. The distribution

of the wave fields over the cross section cannot be described in terms

of simple eigenmode fields, although the poloidal distribution over

any given cross section could be described in terms of an infinite

series of poloidal wavenumbers, m. However, the poloidal spacing of the points intersecting a given cross section varies with toroidal position due to the fact that the helical pitch of a given field

line varies with poloidal angle. Consequently, the spectrum of poloidal wavenumbers describing a natural resonance is not unique.

Similarly, a natural resonance cannot be described in terms of a unique spectrum of toroidal wavenumbers, n. The natural reso.uinces cannot therefore be described in terms of circumferentially periodic toroidal eigenmodes, or found as such solutions of an eigenvalue problem or driven by an eigenmode of the fast wave. If there is no coupling by a fast wave mode, then clearlv there is no mode conversion.

8. ANTENNA COUPLING TO NATURAL RESONANCES

Because the energy flow in the torsional wave is almost totally parallel to the field lines, the Alfven resonant surfaces, and the natural resonances on rational surfaces, are effectively isolated from the rest of the plasma. In order to couple energy into an Alfven resonance surface with an external antenna, and indeed in order to establish conditions for a k = °° resonance, it is necessary to drive the resonance with a fast wave which will propagate across field lines from the antenna. This scheme has been analysed extensively in terms of the eigenmodes of a cylindrical 2b. diffuse pinch, APPERT et al. (1982 a and b, 1983 a and b), ROSS et al.

(1982). In none of these studies, and in none of the studies of

Alfven resonances in toroidal geometry, has it been recognised explicitly that an external antenna will couple not only to fast wave modes but will also couple directly to the torsional wave at all points in the near field of the antenna.

In all practical RF heating experiments, antennas are localized structures. When driven at frequencies less than or near the ion cyclotron frequency, they possess a near field similar to the vacuum field of the antenna. This field differs significantly from the vacuum field only in localized resonance layers or at discrete high frequencies where high Q fast wave cavity eigenmodes are excited (BLACKWELL and CROSS, 1979). In a torus, the fast wave generated by an external antenna propagates isotropically through the plasma and will generate both the near field and one or more cavity modes. A cavity mode can be described by an axial or toroidal wave number k at any given u, with k = n/R. In an inhomogeneous plasma, the wavenumber k for the torsional wave is a locally varying function of position in the plasma at any given w. Consequently, the fast wave can mode convert to the torsional wave at an Alfven resonance layer if the two wave types locally have the same u and k and also have common wave field components. In the far field of the antenna, mode conversion occurs only in isolated Alfven resonant surfaces at any given to. Within the near field of the antenna, fast waves propagate across field lines with a continuous spectrum of k and k. components. Consequently, the fast wave will mode convert to the torsional wave at all points within the near field. The near field of an internal or external antenna can therefore be regarded as a localized per tirbation which generates both the fast and torsional

Alfven wave types, as observed experimentally (lH)R(i ct al., 1984; 27.

CROSS et al., 1982).

Torsional waves generated in the near field of an antenna

will propagate away from the antenna directly along those helical

field lines intersecting the near field. Those rays originating

outside the Alfven resonance layers will propagate independently of

the fast wave since they propagate in a region where the wavenumbers

are locally different and where mode conversion is not possible.

This was the condition under which guided wavepackets were observed

experimentally, as shown in Fig. 4c and as described by BORG et ai.

(1984), although in these experiments the plasma current was zero,

in toroidal current carrying plasmas, rays originating on rational

surfaces will couple to the fast wave at Alfven resonant frequencies

but are decoupled at the natural resonance frequencies. The wave

modes generated by an external antenna will therefore consist not

only of the fast wave modes and the associated k = « Alfven

resonance layers but also the natural resonances due to guided propagation of finite k torsional rays launched by the antenna.

Since natural resonances cannot be described in terms of

conventional toroidal eigenmodes, it is likely that realistic

solutions for any given excitation structure can only be found with

the aid of numerical toroidal codes, such as the ERATO code described by APPERT et al. (1982c). However, such codes have not yet been developed to handle features such as finite frequer.wy effects and wave packet propagation. Some qualitative features can be noted

from the wave field solutions given in equations (7) to (9). With

the aid of (10), the relative magnitudes of the transverse and 2 2 2 2 parallel b field components are given by (bf * b^J/b,, = kj( /F.

At low wave frequencies, P. < 0.2, the b)( component of the torsional wave is negligible compared with the b component, and with the b 28. component when m #0. The b component of the fast wave is also negligible compared with the b and b components when VL < 0.2 for t;he m % 0 surface waves. Consequently, an external antenna, for example a full or half turn loop, will couple to both the torsional wave and the surface waves primarily via the transverse field perturbations induced by the antenna. The transverse field perturbations are strongest near the plasma edge, so that such antennas will couple to natural torsional wave resonances most efficiently at rational surfaces near the plasma edge. At higher wave frequencies, both wave types develop a b.. component which is comparable with the transverse components. Both wave types should be launched more efficiently at higher frequencies, since external poloidal loop antennas produce primarily b field perturbations.

The qualitative features described above are evident in the results described by CROSS et al. (1982). In this experiment, a full turn loop was used in an attempt to excite Alfven resonances in a low temperature (• 20 eV) tokamak discharge. The wave fields were monitored by magnetic probes inserted into the plasma, but to avoid damage to the probes, waves were observed under low current conditions with q(a) ? 9. A multitude of highly localized resonant and anti- resonant surfaces was observed, under conditions where only two or three Alfven resonant surfaces were expected theoretically. The parallel wavelength of the torsional wave in this experiment was shorter than a toroidal circumference on all flux surfaces. These results are consistent with multiple beam interference of rays arriving in phase on the resonant surfaces or out of phase on the anti-resonant surfaces. Small changes in the density and q profiles during the discharge shifted the radial location ol the resonant surfaces, resulting in the appearance of several hundred maxima and zeros in the probe signals during the time history of the discharge. It was 29. also observed that the wave launching efficiency increased with wave frequency and that the wave field amplitude increased with minor radius to a maximum at the plasma edge. These effects were clearly magnetic in origin, not electrostatic, since the antenna was electrostatically shielded and the probe signals were observed to be free of electrostatic pickup.

9. DISCUSSION

Standing Alfven waves on rational surfaces can be driven not only by an external antenna but will also be driven by internal sources of transverse field fluctuations. Such fluctuations are associated with many types of instabilities which are known to occur in ohmicaliy heated plasmas. One would expect that the amplitude of a standing wave on a rational surface under normal discharge conditions will vary statistically with the same thermal noise level as the background fluctuations, but with occasional large excursions due to fluctuations which occur in phase with the standing wave. A sufficiently large excursion may contribute to plasma disruptions as described below. The amplitude of a driven standing wave will also depend on the amount of dispersion and attenuation of a wave packet as it travels around closed field lines.

Dispersion of the moderate and high k .components of an arbitrary torsional wave disturbance is completely negligible under tokamak conditions, even for those components with a perpendicular wavelength \ ** a where a is the plasma radius. The transverse extent of a standing wave near a rational surface is determined not by dispersion of wave energy away from field lines but by the dispersion of energy and loss of coherence associated with the rotational transform of the field lines themselves on surfaces which are nearly but not exactly rational. The transverse dimensions JU. of a standing wave are therefore determined by the q profile and will be typically one or two cm wide in the radial and poloidal directions, representing the typical extent cf spatial coherence.

The maximum transverse dimensions of a standing wave centred on a given q surface is also limited to a few cm by the proximity of adjacent rational q surfaces of low M and N, where q = N/M.

Rational surfaces with high M and N are not expected to support standing waves, since wave attenuation over the length of a closed field line increases exponentially with N.

Attenuation of torsional waves in high temperature plasmas is due primarily to Landau damping. Numerical results for Alfven resonant surfaces in cylindrical plasma* are given by ROSS et al.

(1982) where it is shown that high Q (quality factor) resonant surf-ces occur only near the plasma edge for almost all tokamak conditions. This form of wave damping increases exponentially with temperature up to a maximum when v ~ v. where v is the electron thermal speed. Maximum damping therefore occurs near the high temperature central regions of tokamak discharges and is exponentially weak near the plasma edge. The natural resonance frequency on a rational surfnee in tokamak plasmas is typically of order 1-10 MHz, well below the ion cyclotron frequency, and is relatively independent of plasma radius. Despite the high Alfven speed near the plasma edge where the density is low, the fundamental resonance frequency on a q = 3 edge surface will be approximately the same as that of a q = 1 surface in the centre of the plasma. A high Q resonance is one with a high resonance frequency and weak damping. The most strongly driven natural Alfven resonances in tokamak plasmas will therefore occur on low rational q surfaces near the plasma edge.

Since natural resonances have escaped theoretical and experimental attention for so long, it is natural to enquire about their relevance in regard to the behaviour of tokamak plasmas.

Because of their isolation from neighbouring regions of plasma, and in particular from legions in the shadow of plasma limiters, natural Alfven resonances are unlikely to be detected with external magnetic probes and may only be detected by laser scattering.

Although the torsional wave produces no density fluctuations directly, high k, waves will mode convert to the quasi-electrostatic surface wave near the plasma edge (ROSS et al., 1982, STIX and SWANSON,

1983). There is, however, other indirect evidence which suggests that natural Alfven resonances may play an important role in RF heating experiments and in the stability of tokaraak plasmas.

Most of the present Alfven wave and ion-ion hybrid resonance heating experiments are designed to couple to resonant layers using localised fast wave launching antennas. There will also be direct coupling, by the near fields of such antennas, to the fundamental or harmonics of natural resonances near the plasma edge, particularly with those antennas having strong transverse magnetic field components.

Such resonances will encourage magnetic island formation (see below), especially at high RF power levels, and may therefore contribute to plasma pumpout and impurity contamination commonly observed at the plasma edge in high power RF heating experiments (SHOHET et al., 1979).

Strongly localized Alfven wave heating in rational surfaces will lead to current filamentation as a result of the localized decrease in plasma resistivity and a consequent increase in the local DC current driven by the loop voltago. The rational surfaces are particularly sensitive to minor helical current perturbations.

Such perturbations break up the flux surfaces into magnetic islands, as is a well known consequence of tearing mode theory. It has recently been recognised (CROSS, 1984b) that a sufficiently large current filament on a rational surface, containing only 1 or li

of the total plasma current, can generate a thermally insulating

island surrounding the filament, allowing the filament current to

grow rapidly as a thermal instability. The resulting effects are characteristic of minor or major disruptions, depending on the

initial magnitude of the current in the localized filament. Natural

resonances may therefore contribute directly or indirectly to plasma disruptions. This would be a surprising result, since plasma 2 instabilities are normally regarded as pure CJ < 0 phenomena.

Nevertheless, the effects described above are sufficiently plausible to warrant further numerical and experimental investigation.

Attention has been focussed in this paper on the torsional

Alfven wave, but the slow MHD or ion acoustic wave has many properties in common and is also strongly guided by magnecic fields as shown in lig. 4. This wave is ion Landau damped in collisionless plasmas and has previously been observed only in low density plasmas

(n < 10 ' cm ) with T » T. where Landau damping is weak. The results in Fig. 4 were obtained in a high density plasma, n - 10 cm" under conditions where T and T. were closely equal due to the high collision frequency and where ion Landau damping was ineffective due to the high collision frequency. The question therefore arises as to whether or not ion acoustic waves can propagate in tokamak plasmas where T > T. and where the ion mean free path is typically of the same order as a toroidal circumference. The dispersion relation for ion acoustic waves in a magnetised plasma under these conditions has not been determined, as far as the author is aware. However, the dispersion relation has been determined for an unmagnetiscd plasma, under conditions where the wavelength is comparable to a mean free path (0NO and KULSRUD, 1975). it was found that the phase velocity is approximately the same as that predicted by the fluid model ,}.} • relation and that for T = 2 T., the wave damping or attenuation length would be approximately equal to four wavelengths. Under these conditions, one can expect that ion acoustic standing waves can be driven on rational surfaces in a tokamak, but the quality factor of the resonance would be low. However, a significant reduction in Landau damping occurs as a result of electron drift in a current carrying plasma when the electrons are drifting in the same direction and faster than the wave phase velocity (FRIED and GOULD, 1961). In the case of ion acoustic waves, STIX (1962) has shown that the net Landau damping by ions and electrons is reduced to zero if T. < 0.3 T and if the electron drift velocity l ~ e ' is comparable with the electron thermal velocity. Since these conditions are commonly approacned in tokamak plasmas, ion acoustic waves should propagate with a preferred direction of propagation parallel to the direction of electron drift. Furthermore, natural acoustic resonances may be driven when T. < 0.3 T , as a result of / l ~ e' overstability, by runaway el.-ctrons on or near rational surfaces i (STIX, 1962; AHIIEZER et al., 1975).

If we assume ti.at the phase velocity for the slow M1!D wave is given by the fluid model relation (CROSS, 1984)

a (*) _ S kH [1 • (k^p.)2]* where a is the acoustic speed ~ (kT/m.)* and p. is the ion gyroradius, then a slow MUD wave on a q = 3 surface will have a fundamental resonance frequency of about 10 kHz for typical tokamak conditions

(T - 300 eV) and for k p. < 1. The wave frequency is then comparable with the ion collision frequency and the wavelength is comparable with a mean free path. The frequency also corresponds closely with the frequency of Mirnov oscillations observed in tokamak devices. .>-;.

These oscillations are observed as discrete frequency components of

MUD activity, usually at 10-20 kHz, but as low as 1 kHz in the larger and hotter devices such as PLT and PDX, and as high as 100 kHz in smaller devices such as TOSCA and DIVA. Mirnov oscillations are known to be mode rational and are normally described as being due to rotating magnetic islands generated by tearing modes, but there has been no completely satisfactory explanation for the observed rate of rotation nor any consensus as to whether the observed oscillations are due to poloidal or toroidal island rotation.

It is possible that Mimov oscillations may be associated not with rotation but with wave propagation. An m ^ 0 acoustic mode would have many similarities with the observed features of Mirnov oscillations including apparent rotation, mode rational structure and the presence of discrete harmonic components. The most commonly observed mode of Mirnov oscillations is the [n| = 1, m = 2 mode whose amplitude, observed with magnetic coils near the plasma edge, increases rapidly to a maximum when the q = 3 surface approaches the plasma edge (ClUil-THAM ct al., 1983). The amplitude drops to ~cro if q(a) :> 3.2 or q(a) 4 2.9. From equation (3), this mode could correspond to an acoustic wave with n=-l,m=+2, k=- 1/3R or

A.. = 6TTR localised on and near the q = 3 surface (assuming that

B > 0 and B„ > 0). If the direction of plasma current is reversed

so that Bfl < 0 and q =-3, then the observed mode could correspond to an n = +1, m = +2, k.. = 1/3R acoustic mode. In either case, the

signs of m, n and kl( are consistent with wave packet propagation * along helical field lines in the same direction as the electron drift, the direction of wave propagation being opposite the direction of plasma current toroidally and in the electron diamagnctic drift direction poloidally. These directions correspond to the observed directions of rotation of Mirnov oscillations. There is clearly a .>;>. possible association between Mirnov oscillations and magnetically guided acoustic waves, an effect noted previously in relation to the continuous spectrum by THYACARAJA and IIAAS, 1983.

10. CONCLUSION

A careful examination of the physical properties of torsional

Alfven waves indicates that closed magnetic field lines will support

localised standing waves. Such waves occupy the gap in the continuous frequency spectrum of singular solutions for a torus. The singular solutions, where k . = •*>, are retained formally in the equations used in this paper, but are resolved by considering realistic effects such as finite resistivity, electron mass or finite temperature.

These effects ensure that k, remains finitely large, but the resulting wave types, known respectively as the resistive wave, the quasi-electrostatic surface wave and the kinetic Alfven w.-'vc, still retain the property at moderate to high k, that the group velocity vector is directed primarily parallel to B. Consequently, the basic results of this paper arc expected to remain valid in more realistic plasma models. The effects of plasma current or finite frequency (or any of the other effects just mentioned) are to introduce a small transverse component of the Poynting vector for the torsional wave, but this component is sufficiently small under tokamak conditions that torsional Alfven wave packets will be guided along field lines for many toroidal circumferences. Such waves can be driven, and will propagate independently of the fast

Alfven wave, either by an external antenna or by internal fluctuations in the plasma. Natural torsional wave resonances on rational surfaces may therefore have an important bearing on the behaviour of toroidal plasmas. They may be responsible for plasma pumpout and impurity .1(>.

contamination iu RF heating experiments and may also be important in regard to plasma disruptions. Natural acoustic wave resonances appear to be responsible for some of the observed features of

Mirnov oscillations and may be driven either by runaway electrons on or near rational surfaces or by other M1ID activity near the rational surfaces.

ACKNOWLEDGEMENTS

I would like to thank many people for contributing to this paper including George Collins, Paul Vandenplas, Ian Donnelly, Neil

Cramer, Max Brennan and Robert Winglce. Financial support for this work was provided by ARCS, NERDDC and the Science Foundation for

Physics within the University of Sydney. REFl-RliNCliS

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WINGLEE R.M. (1982) Plasma Physics 24 1161 FIGURli CAPTIONS

Fig. 1(a) Solutions of equation (5) with k = 0, D = 1 m ,

f\ \ R 1 v. =5x10 ms and w . = 10 rs showing the A ci b

torsional (T) and compressional (C) branches for

propagation parallel (k( . > 0) or antiparallel (k .. < 0)

to B. 2 -2 Fig. 1(b) Solutions of equation (S) with kj_ - 50 m for the

same parameters as in Fig. 1(a).

Fig. 2 Variation of normalised phase velocity (oi/k )/v, for

the torsional wave as a function of k , at several

values of D and il with v. = 5 x 10 ms , OJ . = 10 rs" , A CI and k u > 0.

Fig. 3 S /Sj. ratio vs kj_ for k(. > 0 torsional waves at _ , ,,6 -1 , , ^8 -1 v. = 5 x 10 ms and w . = 10 rs A ci

Fig. 4 Oscillograms showing guided wavcpacket propagation for

the acoustic and torsional wave types and isotropic

propagation for the fast Alfven wave.

2 Fig. 5 Variation of k vs minor radius r for parabolic density

and current density profiles showing the torsional (T)

and comprcssional (C) branches for an n = +1, m = +1

mode at SI - 0.1 for typical tokamak conditions. C(k„<0)

- c(k >o)

k»VA

_L 0 0.5 1.0 1.5

-1 Fig. 1(a). Solutions of Eq. (5) with kj_= 0, D = 1 m , vA = 5x10 ms"

Q 1 and LU . = 10 rs showing the torsional (T) and congressional (C) branches for ci propagation parallel (k„>0) or a-itiparaUel(k „< 0) to B.

\ \ 1 \ \ \ \ C(k(|<0)

1.5

C(k)(>0)

II A

T(k„>0)

0.5 1.5

Yiy,. !(!-). Solutions of Ec|. ( O with k £ - >U m " for IIR' b.

II ,4...... !.'.•! :. I •' i Ate. ..'<..<) 1 i I •"•• 1.2 -- -

.: = o.oi D = 0 1.0 : = 0.1 D = 0 - ^^~ - .: = o.i D = 1 0.8 - .. = 0.5 D = 1 1 ^^ .. = 0.5 D ' 3 " A O.o - 1 i I 1

0.-. - .: = 0.9 D = 1 J t 0.1 - y : = o.oi D = 1 1 I i l

1 " 1 1.0 10.0 100.0

l kx(m )

fig. Variation of normalised phase velocity (/kl()/v for the torsional wave 6 -1 ,8 -1 as a function of k. at several values of D and .' with v = 5 y 10 ms , i. . = 10 rs , ci and k '0. 100.0

Fig. 3. S /S ratio vs k^ for k„>0 torsional waves at v - 5 x io'6 -1 X H ms 8 —1 and u) . = 10 rs ci

...... _.., , T T

- kj_ (m ) / C V - J

/ » V / I 1 1 1 1 l

Kig. 5. Variation of k^ vs minor radius r for parabolic density and current density profiles showing the torsional (T) and congressional (C) branches for an n = +1, m = +1 mode at 0.1 for tvpical tokamak conditions. •n>r h- fT7 -*- o." i i:.o .'at cnii.i ir - 01

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!.() era

i- r_ o.:. c;n

2 „s/cra ,,s/cm

lb, I a:'.t M :'.'.-•:. ^.I.'V :i il/..lic;;cii ( I' ;J I .-i <-''• in Ar., jii

Alfvcn • * Acoustic

i | ;i ,; \I: .tii toll .>'>'-<''-1 '•>': i :-n A>.i)ust i c w;r. r i n ilv Ir:'i:cr. i I,. - 1 ••'- '•'• i

V,;\X

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