Theoretical Studies of Low-frequency Shear Alfv´en Waves in Reversed Shear Tokamak Plasmas ∗

The low-frequency Alfv´enic ﬂuctuations in the kinetic thermal-ion gap frequency range have been of research interest since they can interact with both thermal and energetic particles. In this work, linear wave properties of the low-frequency shear Alfv´en waves excited by energetic and/or thermal particles observed in tokamak experiments with reversed magnetic shear are theoretically investigated and delineated in the theoretical framework of the generalized ﬁshbone-like dispersion relation (GFLDR). Since these low-frequency shear Alfv´en waves are closely related to the dedicated experiment of energetic ion-driven low-frequency instabilities conducted on DIII-D in 2019, this work demonstrates, by adopting the representative experimental equilibrium parameters of DIII-D, that the experimentally observed low-frequency modes and beta-induced Alfv´en eigenmodes (BAEs) are, respectively, the reactive-type and dissipative-type unstable modes with dominant Alfv´enic polarization, thus the former being more precisely called low-frequency Alfv´en modes (LFAMs). More speciﬁcally, due to diamagnetic and trapped particle eﬀects, the LFAM can be coupled with the beta-induced Alfv´en-acoustic mode (BAAE) in the low-frequency region (frequency much less than the thermal-ion transit and/or bounce frequency); or with the BAE in the high frequency region (frequency higher than or compara-§ *

The experimentally observed frequencies are also shown.For the BAE, since the modes span a range of frequencies, the lines indicate the upper and lower limits of the unstable bands; for the LFAM, the experimental frequency variation is < 0.5 kHz.In the abscissa, the experimentally measured q min (t) fit shown in Fig. 8 of [34]   is used to convert time to q min , with an associated uncertainty of ∆q min 0.01.In the ordinate, the theoretical labframe frequency incorporates a Doppler shift to the calculated plasma-frame frequency of nf rot , with an associated uncertainty of ∼ 0.5 × n kHz.25, 28, 33, 41] .

[ 34 ]
by Heidbrink et al.The figure shows (a) Cross-power spectrogram in the reference shot for ECE channels between 192-201 cm; (b) measured q min from EFIT reconstructions vs. time.The RSAE( * ), BAE( ), and LFM( ) symbols represent the values of m/n shown on the spectrogram. ã3

Fig. 3 .
Fig. 3.The radial dependences of the typical scale lengths of thermal and energetic particle pressure (L P th and L P E ), magnetic shear (s) as well as the estimated radial mode width (∆ m ).

Fig. 4 .
Fig. 4. The radial dependence of the normalized pressure gradient of EPs with the classical profile.Here, the normalized radial position of q min is ρ 0 ≡ r 0 /a = 0.28.

3Fig. 5 .
Fig. 5. Radial profiles of (a) temperature and q, (b) density and (c) B t as well as toroidal rotation frequency f rot of DIII-D shot #178631 used for numerical studies.

Fig. 7 .Fig. 8 .
Fig. 7. Dependence of the (a) real frequencies, (b) growth rates and (c) polarization of the low-frequency SAWs on Ω * pi ≡ ω * pi /ω ti for the cases without (w/o) and with (w/) EP effects.Here, a dashed vertical line represents the experimental value of Ω * pi;exp of about 0.35.

45 keVFig. 10 .
Fig. 10.The dependence of mode frequency and growth rate on T e .Here, T i = 2.45 keV.