TAE INTERACTION WITH ENERGETIC PARTICLES AND THERMAL PLASMA, TAE STABILITY
Now we consider how the particle-to-wave power transfer works for TAE in the presence of charged energetic particle population. First, we clarify the components of the field perturbations associated with TAE, for which solutions (6.15), (6.16), and (6.20), (6.21) were found. The incompressible ideal MHD approach used for TAE above allows neither parallel electric nor parallel magnetic perturbations, <5£ii =0, (5/in = 0. The shear Alfven waves with perturbed scalar ip and vector 8A potentials satisfy then:
We express perturbed /и-th perpendicular electric and magnetic fields via (pm:
We introduce an additional factor corresponding to an exponential growth or damping rate of the mode amplitude due to the particle-to-wave power transfer at an early linear phase of the wave- particle interaction,
The net exponential growth/damping rate of TAE amplitude is determined by
where Pa is the power transfer from charged energetic particles to the wave, P(l is the power absorbed by the background plasma, and (5IT is the wave energy given by the sum of field energy and kinetic energy due to 8E x B0 drift:
The mode is linearly stable when у < 0.
Power Transfer from Energetic Particles to TAE
For considering energetic particle drive, we take an example of burning plasma in ITER baseline scenario with current IP= 15 MA, R0 = 621cm, u= 200cm, and B0 = 5.3 T, ne(0) ~ 102onT3. For TAE, 1 MeV deuterium beam ions and alpha particles we obtain VA=1- 108cm/s < УЫат (f = 0) = 109cm / s < V„(f = 0) = 1.3 • 109cm/s. so that both types of the energetic ions during their slowing-down process could enter resonance Vn = VA.
Because TAE has no parallel electric field, 8En = 0, the power transfer from a resonant alpha particle to the wave comes in the guiding centre approach from the alpha particle orbit drifting across the TAE magnetic flux surface:
where the unperturbed guiding centre drift velocity is
Figure 6.5 shows a two-dimensional projection of passing or trapped drift resonant alpha particle orbit moving from point A to point В across the radial structure of TAE mode with electric field SE.
When a resonant ion moves radially across TAE from point A to point B, it crosses the electrostatic potential associated with TAE. The mode and the ion exchange energy eA(p during the radial drift motion is shown in Figure 6.5.
We can see here that the particle-to-wave power transfer arises because TAE is attached to magnetic flux surface, while the drift orbit moves across in radius and is displaced from the magnetic surface. Furthermore, it is easy to understand from Figure 6.6 that the particle-to-wave power transfer depends on the ratio between the radial width of the orbit and the radial width of TAE. In fact,
FIGURE 6.5 Schematic geometry of the interacting TAE attached to the magnetic flux surface, and passing (left) or trapped (right) drift orbits of the interacting resonant particles displaced from the magnetic surfaces.
FIGURE 6.6 Schematic radial projection of the two-dimensional plot in Figure 6.5 showing the drift orbit crossing one harmonic of the electrostatic potential associated with TAE.
there are two characteristic radial widths of TAE. The one determined by the mode structure in the inner layer (6.15) and (6.16) where toroidal coupling matters is
The second characteristic TAE width is determined by the outer mode structure (Eqs. 6.20 and 6.21):
Figure 6.7 shows qualitatively how the growth rate of energetic particle-driven TAE increases linearly at small Д0 / Д„, = A0 ■ m / rTAE corresponding to the A,-B, orbit in Figure 6.6. The growth
FIGURE 6.7 Qualitative dependence of the energetic particle drive for TAE as a function of the ratio between the energetic particle drift orbit width and the radial width of TAE.
rate saturates when the orbit becomes comparable to the inner width of TAE [6.19], and decreases at the orbit size much larger than the outer width of TAE (this case corresponds to the A2-B2 orbit in Figure 6.5 [6.11]). The highest energetic ion drive is achieved at
when the maximum is achieved in the value |c«A
We can extend the particle-to-wave power transfer from the single particle consideration to a distribution function of energetic particles. The particle-to-wave power transfer for the whole population of energetic particles takes the form
where / is the linear perturbed distribution function of energetic particles.
For calculating (6.47), we assume for simplicity that [6.19]
and consider passing ions for which 6 = vц / R,i? = vM / qR. The drift kinetic equation for energetic ions takes the following form:
where Ч1 = пф- mi? - cot.
To represent the electrostatic potential (pm of TAE in the reference frame associated with the drift orbits of passing ions, we change variables (r, i?) —> (r,i?) meaning the transform to the orbit reference frame:
This change in the variables gives so Eq. (6.49) transforms to
where the electrostatic potential of TAE seen by the energetic ion along its orbit depends on the poloidal angle i) and can be represented as a Fourier series in i?:
The wave-particle resonance condition for the passing ions has the form so the resonance contribution from the kinetic equation (6.52) takes the form:
Integration of (6.47) for this perturbed distribution function and for strongly passing, — = 1, beam distribution function f0 gives
where the finite drift orbit parameter D is given by
estimated at the resonant velocity V[, = VA, and the coupling integral is shown in Figure 6.8.
Note that the resonance condition (6.54) takes the following form for TAE:
where / = integer describes the wave-particle resonances. In particular, the principal resonance,
corresponds to / = 0,
and the strongest side-band resonance,
corresponds to l = -1.
FIGURE 6.8 Coupling integral for passing beam particle principal resonances: (a) asymptotic expression (broken curve) and numerical result (full curve); (b) function 1(D)ID used in Eq. (6.55).
For isotropic distribution function of fusion-generated alpha particles, the alpha particle drive for TAE is estimated to be
where the finite drift orbit parameter D for passing alpha particles resonating with TAE via the Vp = VA resonance is given by
The drive theory could be extended to a general case of wave-particle resonances including trapped particles and the distribution with gradients in both velocity space and in the real space. The description of a guiding centre distribution function in a five-dimensional phase space could be significantly simplified with a proper choice of variables. In tokamaks with toroidally axisymmetric magnetic field, three invariants of the unperturbed particle orbits are conserved: particle energy E = maV2 / 2, magnetic moment /и = maV2 / (21?), and the toroidal angular momentum,
where T'(r) is the poloidal flux, V,V±, Уф are total, perpendicular (to the magnetic field), and toroidal velocities of the particle, ea,ma are the charge and mass of the particle, and R is the major radius. It is convenient to use the variables (E, P0.p) for describing the energetic particle distribution function. fa(E, P0, /и), and the characteristic toroidal, 0)0 (£.1^,//), and poloidal, <^>e(E.P0.p), orbit frequencies of the particles. The resonance condition between TAE with frequency со and toroidal mode number n and the energetic particles takes the following form in the general case:
The wave growth rate in the general form has the contributions from the gradients of the energetic particle distribution in both radial and energy space:
TAE Damping due to Thermal Ions
Due to the side-band resonance (6.59), the exponential tail of the Maxwellian distribution of thermal
ions may interact with TAE. The efficiency of the resonance Vy = — is significantly lower than the efficiency of the principal resonance Vf, = VA between TAE and energetic ions. However, the density of thermal ions exceeds density of energetic ions by several orders of magnitude, and hence, the power transfer between thermal ions and TAE may be comparable to that between energetic ions and TAE [6.20].
Thermal ion species burning ITER plasmas are of four main types:
where mH and eH are the mass and charge of hydrogen ion (proton). For these ions, the quasineutrality condition is:
and the plasma depletion due to the impurity ions is as follows:
Here, we consider the depletion factor in the range
where the lower boundary is determined by Be impurity ions coming from Be wall in ITER and always present in the plasma at the level of -2%. Another major contributor to the depletion is He ash population, the density of which is determined by the intensity of DT fusion and the efficiency of He ash removal by He transport and pump-out. The D:T mixture in burning plasmas is supposed to be close to 50:50, so we assume
It follows then that the ion densities are related to the electron density via:
and Be density is 2%. The value of Alfven velocity in such plasma is
showing a higher sensitivity to the plasma depletion than to the deviation in the D:T mixture from the optimum value 50:50.
The drift frequency of thermal ions is much lower than TAE frequency, ы,, / ft)TAE| « 1, so we can neglect the drift ion effects and consider only the contribution due to the term codFj / ЭE in (6.64) to obtain the estimate of thermal Dion Landau damping [6.15]:
Here, the term with rD = Te / 7b is caused by finite parallel electric field of TAE, and the plasma was assumed with D ions having the Maxwellian distribution function and thermal velocity ^td:
Note here that the ion Landau damping does not depend on the mode number.
For the plasma consisting of a D-T mix of ions the ion Landau damping is a sum of the contributions from D ions and T ions,
where damping due to D ions is determined by (6.71) with D ion density and temperature, and yT is given by a similar expression with index T instead of D. Combining the damping effects due to D and T thermal ions gives
so the relative contribution of T ions could be easily assessed.
TAE Damping due to Thermal Electrons
For ITER baseline scenario, we obtain the following ordering:
On the other hand, for an effective electron Landau damping, the principal resonance condition should be met,
Combining (6.75) and (6.76), we obtain for thermal electrons
This inequality could only be possible if
that is, the majority of electrons satisfying (6.76) are trapped. For trapped electrons, the main mechanism of electron damping is associated with Coulomb scattering of the electrons near the trapped- passing boundary [6.14,6.17,6.21], which gives
This damping effect depends on the /и-number (via ДТАЕ). In particular, this effect could effectively absorb outgoing KAW waves when kr —>
Although the electron Landau damping of TAE is small at the position of the TAE-gap where the mode phase velocity is Alfven velocity, the phase velocity, co/kllm, can increase significantly away from the TAE-gap as the eigenfrequency is fixed while the parallel wave-vector depends on the radius, kW = klln, (r). Electron Landau damping may contribute then [6.22], but it affects the tails of the mode structure (6.24), which are small.
Continuum Damping of TAE
Finally, some continuum damping of TAE may still exist if the TAE frequency crosses the Alfven continuum at a certain point in radius. This damping could vary from zero in the case of the so- called “open” TAE-gap (no crossing exists) to a high value when the TAE-gap is closed and the continuum crossing point in radius is close to the TAE-gap. This effect was experimentally validated on JET, where the TAE damping measured with the external TAE antenna varied from 0.6% in the open TAE-gap case, to a very high value of 5% in the case of a closed TAE-gap [6.23]. In a limiting case of very high mode numbers and when the TAE-gap is closed, the expression for the continuum damping was obtained in Refs. [6.24,6.25]:
A.B. Mikhailovskii, Sov. Phys. JETP 41 (1975) 980. 2 M.N. Rosenbluth and P.H. Rutherford. Phys. Rev. Lett. 34 (1975) 1428. 3. G.Y. Fu, PhD Thesis, The University of Texas at Austin (1988). 4. C.Z. Cheng et al., Ami. Phys. (N.Y.) 161 (1984) 21. 5. G.Y. Fu and J.W. Van Dam.. Phys. Fluids B1 (1989) 1919. 6. L.Chen, Theory of fusion plasma (Proc. Varenna Int. Workshop), edited by J. Vaclavik, F. Troyon, andE. Sindoni, Societa Italiana di Fisica-Editrice Compositori, Bologna (1989) p.327. 7. K.L. Wong et al., Phys. Rev. Lett. 66 (1991) 1874. 8. W.W. Heidbrink et al.. Nucl. Fusion 31 (1991) 1635. 9. R. Nazikian et al.. Phys. Rev. Lett. 78 (1997) 2976. 10. H.L. Berk et al.. Phys. Fluids B4 (1992) 1806.}} 
- 15. J.W. Connor et al., Non-ideal Effects on TAE Stability, Proceed, of the 21st EPS Conference on Control. Fusion and Plasma Phys., edited by E. Joffrin. P. Platz, and PE. Stott, (Montpellier, Vol. 18B, 27 June 1994), p. 616.
- 16. R. Nyqvist and S.E. Sharapov, Phys. Plasmas 19 (2012) 082517.
- 17. J. Candy and M.N. Rosenbluth. Phys. Plasmas 1 (1994) 356.
- 18. J.W. Connor et al., Phys. Fluids 26 (1983)158.
- 19. H.L. Berk et al., Phys. Lett. A162 (1992) 475.
- 20. R. Betti and J.P. Freidberg, Phys. Fluids B3 (1991) 1865.
- 21. N.N. Gorelenkov and S.E. Sharapov, Physica Scripta 45 (1992) 163.
- 22. J. Candy, Plasma Phys. Control. Fusion 38 (1996) 795.
- 23. A. Fasoli et al., Phys. Rev. Lett. 75 (1995) 645.
- 24. M.N. Rosenbluth et al., Phys. Rev. Lett. 68 (1992) 596.
- 25. F. Zonca and L. Chen, Phys. Rev. Lett. 68 (1992) 592.
-  B.N. Breizman and S.E. Sharapov. Plasma Phys. Control. Fusion 37 (1995) 1057. 12. A.B. Mikhailovskii and S.E. Sharapov, Spectrum of Ideal Ellipticity-Induced Alfven Eigenmodes in aLow-Shear Tokamak, Proceed, of the 5th IAEA TCM on Alpha Particles in Fusion Research (JET JointUndertaking, Abingdon. Oxfordshire, UK 1997). 13. R.R. Mett and S.M. Mahajan, Phys. Fluids B4 (1992) 2885. 14. J. Candy and M.N. Rosenbluth, Nucl. Fusion 35 (1995) 1069.