The ideal MHD approach gives the following equation for shear Alfven waves describing a balance between the bending energy term (first term in the equation) and the inertial term (second term in the equation):

Here, the plasma pressure gradient is assumed to be negligibly small, ^ S2- The derivation of (6.1) is given in Appendix C. To analytically investigate this three-dimensional equation, we transform it to a set of one-dimensional coupled equations by following the approach developed in Ref. [6.10]. This is done by taking into account that the wave solutions in bounded toroidal plasmas should be periodic and quantised in toroidal and poloidal directions:

where n is the number of wavelengths in toroidal direction, m is the number of wavelengths in poloidal direction, and c.c. stands for “complex conjugate.”

By assuming concentric magnetic flux surfaces, low magnetic shear, and taking only the first- order inverse aspect ratio expansion, —

mode, m-the and (m- l)-th, and obtain (see Appendix C) two coupled second-order differential equations for the amplitudes (p„, (r), m_, (r) of the wave electrostatic potential:

where the toroidicity coupling coefficient is £ = ^ —j, and the differential operator L„, is defined as

The set of Eqs. (6.3)—(6.5) describes TAEs.

In toroidal geometry with plasma current, the parallel wave-vector of the m-th harmonic of a wave with toroidal mode number n has the form

and its radial dependence is determined by the safety factor q(r) = rB$ / RBt). Two important properties of the parallel wave-vector are essential for further analyses. First, the parallel wave- vector (6.6) could be zero, in contrast to the cylindrical geometry. Indeed, if for some values of q (taken at a certain radius r), and integers n, m the condition m = nq is valid, a rational magnetic surface with k,„ = 0 exists at this radius. This implies that the frequency of the shear Alfven continuum starts from zero, in contrast to the cylindrical plasma case. Second, two parallel wave- vectors of neighbouring poloidal mode numbers, m and m -1, could be equal at the same radial position satisfying

thus making the continuum spectrum degenerate. Let us explain this specific property of toroidal geometry in detail.

In cylindrical geometry,--» 0, two poloidal modes (p,„ (r) and m. (r) are decoupled, as shown


in Figure 6.1 (left) with two broken lines. Equations (6.3) and (6.4) become singular at kl„Vl and a>l = kl„.Vx, which give the two cylindrical shear Alfven continua. In toroidal geometry, Eqs. (6.3) and (6.4) become coupled due to the finite toroidicity, as the poloidal mode numbers are not good quantum numbers any longer. The shear Alfven continuum in toroidal geometry is obtained by setting the determinant of the coefficients in front of the second-order derivatives in (6.3), (6.4) equal to zero, that is,

This gives the following two branches:

At the point where the two cylindrical continua would cross, a gap in the toroidal shear Alfven continuum appears with the frequency width

The toroidicity-induced gap in the frequency of shear Alfven continuum has two extrema, at the bottom and the top of the gap, satisfying

Left: Radial structure of two cylindrical Alfven continue in toroidal plasma. Right

FIGURE 6.1 Left: Radial structure of two cylindrical Alfven continue in toroidal plasma. Right: two poloi- dal harmonics form an eigenmode localised around radius satisfying Eq. (6.9), with the eigenmode frequency inside the TAE-gap.

similar to the case of GAE in cylindrical plasma with current. As in the case of GAE, we can expect that the toroidicity-induced extrema (6.12) and the corresponding extrema in the perpendicular refraction index could cause a wave-guide at the radius r0, where an eigenmode may be formed.

To find discrete TAE spectrum from Eqs. (6.3) and (6.4), we introduce a normalised frequency as follows:

so this frequency is g = 0 at the centre of the TAE-gap where the two cylindrical continua cross, (6.10), and g = ±1 at the bottom and the top of the TAE-gap. We will now look for a discrete TAE eigenfrequency inside the TAE-gap frequency. In the case of a low magnetic shear, such frequency is close to the bottom tip of the continuum [6.11]

(right) explains the structure of the gap in the normalised frequency units

Figure 6.2 (right) explains the structure of the gap in the normalised frequency units.

Next, we will consider two radial regions of Eqs. (6.3) and (6.4): the inner region in the vicinity of the extremum point determined by (6.10), and the outer region away from (6.10). In the inner region, the effect of toroidicity is essential and the radial derivatives exceed significantly the poloidal scale

of the wave variation, — » —, so that last term in (6.5) could be neglected. Eqs. (6.3) and (6.4) dr r

can be integrated in the inner region to give Left: Radial structure of two coupled poloidal harmonics of TAE. Right

FIGURE 6.2 Left: Radial structure of two coupled poloidal harmonics of TAE. Right: the normalised frequency g in the vicinity of a TAE-gap.


and C,„, C,„_i are the integration constants.

The symmetric (with respect to z —> -z transform) logarithmic parts of the inner solutions (6.15) are the remnants of the logarithmic solutions that gave the singularity at the continuum resonance in cylindrical plasmas. Now, there is no logarithmic singularity in toroidal geometry because of the choice of the eigenfrequency (6.14) inside the gap in the continuum.

The asymmetric parts of the type tan"1 (a-z) of the inner solutions are caused by the involvement of every poloidal harmonic in the radial regions at to two neighbouring gaps. These asymmetric parts give jumps of the inner solutions:

These jumps should be compensated by the outer parts of the solutions as there must be no asymmetry in the TAE electrostatic potential at z —> ±~.

The outer solution describes the “cylindrical” harmonics away from the region of toroidal coupling. The outer solution is obtained from Eqs. (6.3)—(6.5) by neglecting toroidicity and taking

d m .

---, which gives

dr r e

where x = (e / 4)z.

The outer localised solutions of Eqs. (6.18) and (6.19) are:

where Dm, Dm ., are constants, and U(a, b, cx) is the confluent hypergeometric function. For matching the inner layer solutions, we take the asymptotics of (6.20), (6.21) for |;c|/.S 1:

where у is the Euler constant.

In the limit of low magnetic shear, S «: 1, the confluent hypergeometric function can be expressed via the Bessel function:

The jumps in the inner layer, (6.17), compensated by the jumps in the outer region, (6.22) and (6.23), give the dispersion relation for TAE with the same sign of two poloidal harmonics, that is, Cm ~ Cnl-[.

which transforms to TAE frequency by taking into account Eqs.(6.13) and (6.14):

In plasmas with ellipticity and triangularity of the cross-sections, double and triple oscillations in the poloidal angle appear in the components of the metric tensor (these oscillations were not considered in Appendix B). The double and triple oscillations give rise to other types of the “gap” modes in the shear Alfven continuum in addition to the toroidicity-induced gap. Namely, ellipticity-induced gaps and ellipticity-induced Alfven eigenmodes (EAEs) exist due to the coupling between poloidal w-th and (m + 2)-th harmonics, and triangularity-induced gaps and non-circular (triangular) Alfven eigenmodes (NAEs) exist due to the coupling between poloidal m-th and (m + 3)-th harmonics. Figure 6.3a shows an example of a computed spectrum of waves in a typical JET plasma w'ith ellipticity and some triangularity, with compressibility effects taken into account. The Alfven spectrum is a mixture of continuum bands and gaps in this continuum, inside which discrete eigenfrequen- cies exist corresponding to TAEs and EAEs. At low frequency, a computed ion sound continuum is seen. Figure 6.3b show's that computed radial structure of TAE in the plasma with elliptical cross-section adds more poloidal harmonics. An analytical theory of EAE in a low-shear tokamak

(a) Left

FIGURE 6.3 (a) Left: Radial structure of shear Alfven continuum with n = 1, Im(A) = Im(yR0 / VA (0)), in a typical JET discharge. Right: the shear Alfven spectrum corresponding to Re(X) = 0 (waves, no instabilities), (b) Radial structure of n = 1 toroidal Alfven eigenmode continuum with frequency marked by the arrow in (a). In addition to the two dominant poloidal harmonics of m = 1 and m = 2, higher poloidal harmonics are also coupled due to the large magnetic shear and non-circular plasma cross-section.

was developed similar to TAE. with the ordering — » — and tw'o starting equations for the ampli-

dr r

tudes (pm (r), (pm-2 (r) of the electrostatic potential [6.12]:

where the ellipticity coupling coefficient is e =2 - l) / (£2 + l), £ = b / a, and a, b are the lengths of the elliptical semi-axes.

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