Evolution of Alfven Continuum due to Temporal Evolution of Non-Monotonic q-Profile

Because the ACs exist in reversed magnetic shear discharges and seem to be localised in the vicinity of the min magnetic flux surface, an investigation of Alfven waves associated with the region surrounding the zero magnetic shear point, as well as study of the temporal evolution of the key equilibrium parameters in this region are necessary for interpreting ACs. We start by investigating the Alfven continuum in the toroidal geometry and for plasma equilibrium with non-monotonic «/-profile. Due to the non-monotonic «/-profile, the shear Alfven continuous frequency as a function of radius has extremum points, r=r₀, satisfying

and so the Alfven spectrum may contain discrete eigenvalues as in the cases of GAE and TAE.

The «/-profile evolves during AC observations, so we need to investigate the Alfven continuum as a function of radius and time. We calculate the continuum wuth the ideal MHD CSCAS code, which accounts for toroidal geometry. The temporal evolution of the Alfven continuum frequency at the point of zero magnetic shear at q_min (“tip” of the Alfven continuum) is of major interest. In a “cylindrical” limit, this evolution is described by:

where r_min is the zero magnetic shear point, and q_min(t) and V_A(t) vary in time in accordance with the experiment. This behaviour is quite similar to what emerges from the toroidal CSCAS code except that the numerical code automatically switches the dominant /«-number as the frequency approaches the TAE-gap frequency. Figure 9.8 shows the evolution from the CSCAS code of the value w_A(t) for« = 1 as q_mm(t) gradually decreases from 3 to 2.4. It follows from Figure 9.8 that during the entire evolution, the Alfven continuum frequency as a function of radius has a local extremum point determined by (9.2), which is very close to the point of zero magnetic shear, q_min. During most of this time evolution, the Alfven continuum has a local maximum at ^_min except near the final point where q_mm approaches <7,_nin=2.4 (labelled 7). At low frequency, this local maximum of the frequency linearly increases as ^_min decreases, but as the maximum frequency approaches the TAE-gap, the local maximum changes to the local minimum of the Alfven continuum at q_min (marker 7 in Figure 9.8). During the subsequent decrease of q_min to 2, this minimum of the Alfven continuum decreases to zero frequency as the rational magnetic surface is formed at q_min= 2. On a longer time scale, the evolution of « = 1 Alfven continuum frequency as a function of q_min is shown in Figure 9.9 [9.2].

The repetition rate of the Alfven continuum tip passing zero frequency is determined by the rate of plasma current increase and diffusion, which causes q_min to pass integer values during the interval when q_min decreases. During this scan, the highest frequency achieved by the local maximum of the

FIGURE 9.8 The CSCAS modelling: normalised frequency (dRJV_a{0) of n= 1 Alfven continuum as a function of radius 5 = (y/> / ty/p) for several values of q_mm associated with evolution of q_mm (/) from q_mm=3 down to <7_mul=2.4 in reversed-shear discharge. The sequence of Alfven continuum “tips” corresponding to values q_min =3, 2.9, 2.8,..., 2.4 is shown by numbers 1,..., 7.

FIGURE 9.9 The CSCAS analysis showing temporal evolution of the normalised frequency / V_A

at q = q_min as q_nm(t) decreases in time. Mode numbers plotted are n = 1. n = 2, and n = 3. Solid lines indicate times of local maximum of the Alfven continuum, while broken lines indicate times of local minimum of the Alfven continuum.

Alfven continuum is bounded by the frequency of the TAE-gap. Because the frequency of TAE-gap is inversely proportional to q_min, the envelope of the highest frequencies is roughly proportional to the plasma current increase during the time of observation. If now we consider the temporal evolution of the Alfven continuum frequency at q_mi„ for higher toroidal mode numbers, n=2 and n=3, we observe a similar characteristic pattern of the intermittent maximum and minimum of Alfven continuum at decreasing q_mm. However, the rate of the frequency sweeping and the repetition rate for the higher mode numbers are higher than for n= 1, for example, the n=2 mode frequency sweeps approximately twice as fast, and this mode passes zero frequency not only at integer values of q_mm but at half-integer values as well.

A direct comparison of the experimental data shown in Figure 9.4b and the CSCAS modelling in Figure 9.8 suggests that the following formula describes the frequency sweeping of the Alfven cascades:

where Aw is an off-set frequency determined by possible Doppler shift in toroidally rotating plasma, by toroidal coupling corrections of the Alfven continuum, etc.

Taking into account the correspondence between the temporal evolution of the Alfven continuum tip and the experimentally observed data, our basic hypothesis concerning ACs is that these modes are similar to a global Alfven eigenmode, whose frequency is close to the frequency where the localised shear Alfven wave has an extremum as a function of radius. However, in the standard theory of the GAE, in cylindrical geometry, the extremum of the local Alfven wave is a minimum. However, we see on comparing Figure 9.4b and 9.8 that only upward sweeping ACs are observed experimentally, which correspond to the maximum in Alfven continuum, in contrast to GAE.

Theoretical Explanation of ACs

Taking into account all the essential results of our previous analysis and the experimental data, a discrete spectrum of modes associated with non-monotonic ^(r)-profiles was found in Refs.

[9.2,9.13]. Within the MHD description of shear Alfven perturbations and the drift kinetic description of energetic particles, the following equation was derived for a single cylindrical harmonic of the mode electrostatic potential:

Here, the left-hand side consists of the usual shear Alfven terms with Alfven velocity calculated for a local mass density but for the on-axis magnetic field. The right-hand side represents two different effects causing the formation of an AC eigenmode: the first term - e² results from the toroidal geometry and is important for plasmas with weak reversed shear, while the terms in the second bracket with flux-averaged density and current of energetic (hot) ions result from electrons compensating energetic ion charge when drift orbits of energetic ions are much larger than the mode width. The very large orbit width compared to the radial mode width is clearly visible on comparing Figures 2.7 and 9.7. By looking for an eigenmode with eigenfrequency very close to the Alfven continuum “tip” frequency (9.3) in the vicinity of q_min and taking into account the zero shear condition at the position r = r₀ of q_mm, we expand the parallel wave-vector around <7,ni„ (to) =

while the frequency is supposed to be close to ©„ = h (r₀ )-V_A (r₀):

Then, the starting Eq. (9.5) reduces to the form

where

This second-order differential equation can be represented in the form of Schrodinger equation [9.2,9.13], with the existence of a discrete eigenvalue (corresponding to the AC mode existence) possible if

FIGURE 9.10 Schematic plot of normalised AC frequencies £2 = co/(Otae for two selected poloidal mode numbers M and M-1. The different quadrants correspond to different signs of <2_hot and g_lor. Solid lines indicate possible AC modes, dashed lines indicate the Alfven continuum, the circles indicate the TAE frequency regions where the single harmonic mode approximation fails. The relevant values of in are shown in each quadrant*.

Reproduced from [B.N. Breizman et al., Phys. Plasmas 10 (2003) 3649], with the permission of AIP Publishing.

where

Figure 9.10 shows schematically what types of ACs may exist below and above TAE-gap depending on the contributions from (9.11) and (9.12). These values could be positive or negative depending on the value of q₀ and the sign of AC frequency.

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