Several research groups investigated possible heating of plasma with externally launched waves of the SA frequency range. It was found in numerical modelling that in cylindrical plasmas, as shown in Figure 5.4, in addition to the continuous Alfven spectrum,

A new high-quality, *Q = w/y~* 10 resonance was discovered during these Alfven antenna studies in plasmas with current, as shown in Figure 5.5.

FIGURE 5.4 Geometry of the plasma, coil, and the wall cylindrical system used in the numerical investigation*.

FIGURE 5.5 Real part of the computed coil impedance versus normalized frequency ft)/(fc”^{m} *- V*_{A}). A new resonance with a very high quality is detected just below the continuum frequency in plasma with current*.

Reproduced from [D.W. Ross et al. *Phxs. Fluids 25* (1982) 652], with the permission of AIP Publishing.

The theoretical interpretation of the new discrete frequency GAE was provided. It was shown that GAE exists in ideal MHD if the current profile determining d*B*_{t)}/dr provides a minimum in Alfven continuum, that is,

which can be expressed as a condition for parallel wave vector at a given plasma density profile:

Figure 5.6 shows that all continuum curves for various poloidal mode numbers had minima in the modelling that included plasma current.

The local minimum of Alfven continuum provides a maximum of the perpendicular refraction index *N*_{r} = ck_{r}/eo that describes electromagnetic waves (and SA wave is an electromagnetic wave). In the presence of the maxima in the refraction index, similar to the fibre optics, the wave tends to propagate in a “wave-guide” surrounding the region of the extremum refraction index. Figure 5.7 shows that the structure of GAE peaks in the vicinity of the extremum point satisfying (5.31). Furthermore, we see no logarithmic singularity of the type (5.28), (5.29) in the computed GAE. This means that although GAE is a SA perturbation, it does not experience the continuum damping. The most significant damping typical of the SA wave packets satisfying (5.19), does not exist for GAE as its eigenfrequency does not cross the local Alfven frequency anywhere in the plasma. Because GAE frequency does not satisfy the local Alfven resonance condition,

this SA mode has no singularity, does not experience the continuum damping, and belongs to weakly damped Alfven Eigenmodes (AEs) with high quality factors.

FIGURE 5.6 Radial structure of Alfven continuum in cylindrical plasmas with current. Poloidal mode numbers from -1 to -3 have minimum points.

FIGURE 5.7 Radial structure of the computed ideal MHD GAE with *m=—2.*

# KINETIC ALFVEN WAVES VERSUS SHEAR ALFVEN WAVES

Kinetic Alfven waves (KAWs) [5.5] represent the most important branch of waves that can couple with SA waves in inhomogeneous plasmas, and which often determine the microscopic mechanisms of the damping of SA waves. KAWs are associated with finite Larmor radius effects, which play an important role when the perpendicular wavelength, *2n/k*_{±}, becomes comparable to the Larmor radius of thermal ions. In such short wavelength perturbations, ions can no longer follow the magnetic field lines, whereas electrons are still frozen in the field lines because of their small Larmor radius. This produces charge separation and additional parallel electric field.

In particular, the continuum damping of SA wave, which results from the singularity in the wave amplitude at the resonance position (5.23) in ideal MHD in real plasma corresponds to the wave transformation into KAW. Figure 5.8 shows a comparison of the mode structures near Alfven resonance position for the ideal MHD approximation and KAW. In this figure, an SA wave launched from the left side approaches the Alfven resonance position at *X=0.* As it does so, the wave amplitude increases and gives in ideal MHD model a singularity and the wave continuum damping in accordance with (5.28) and (5.29). In real plasma, KAWs are excited as soon as the distance between the wave launched and the resonance point becomes comparable to ion Larmor radius. The wavelength of KAW is of the order of the Larmor radius, and the wave has an oscillatory structure propagating across the magnetic field.

The dispersion relation of KAW has the form

In contrast to SA wave, KAW can propagate across the magnetic field, and has electric field components not only across the magnetic field but also in the direction of the magnetic field. Note also that this wave propagating through a region with *k*_{n} = 0 (e.g., across rational magnetic surfaces in tokamaks) becomes of an infinitely short wavelength, *k*_{L} —> <*>, since its frequency is constant. This makes KAW modelling difficult in real space, unless an absorption of some kind increasing with *k*_{L }exists for the wave, which can cut off the very short wavelengths.

FIGURE 5.8 Structure of kinetic Alfven wave in the vicinity of the Alfven resonance.

Finally, we note that in plasmas with ordering V_{Te} s *V*_{A}, which is rare in tokamaks, the electron inertia becomes important and the KAW dispersion relation is modified to

# REFERENCES

B.B. Kadomtsev, Kollektivnye yavleniia vplazme, NAUKA, Moscow (1976) (in Russian). 2. A. Hasegawa and C. Uberoi, The Alfven Wave. A.Hasegawa and C.Uberoi. Technical InformationCenter, U.S. Department of Energy, (1982). 3. D.W. Ross et al. Phys. Fluids 25 0982) 652. 4. K. Appert et al. Plasma Phys. 24 (1982) 1147. 5. A. Hasegawa and L. Chen, Phys. Rev. Lett. 32 (1974) 454.