Alfvén Acoustic Channel for Ion Energy in High-Beta Tokamak Plasmas

When the plasma beta (ratio of thermal to magnetic pressure) in the core of a tokamak is raised to values of several percent, as required for a thermonuclear fusion reactor, continuous spectra of long-wavelength slow magnetosonic waves enter the frequency band occupied by continuous spectra of shear Alfvén waves. It is found that these two branches can couple strongly, so that Alfvén modes that are resonantly driven by suprathermal ions transfer some of their energy to sound waves. Since sound waves are heavily damped by thermal ion Landau resonances, these results reveal a new energy channel that contributes to the damping of Alfvénic instabilities and the noncollisional heating of bulk ions, with potentially important consequences for confinement and fusion performance.

Figure 1

MHD equilibrium. (a) Pressure contours. (b) Radial profiles of the safety factor $q(r)$ and plasma beta $\beta (r)$.

Figure 2

Spectrograms of MHD responses for ${\beta }_0=3.6\%$ and $n=3$, computed using Eq. (2) with $t_{\text{win}}=1100$ for (a) the toroidal $\delta u_{\zeta }$ and (b) radial velocity component $\delta u_r$. The color scale is logarithmic, with strongly responsive regions (or slowly decaying modes) appearing red to orange. Lines and dots indicate full MHD continua in (a) and SAW continua in (b).

Figure 3

Comparison of $n=3$ MHD responses for ${\beta }_0=1.7\%$ and 3.6%. (a),(b) Contours of spectrograms $|\delta u_r|(r,\omega )$ with $t_{\text{win}}=1100$. The contours above and below the dash-dotted lines have independent linear color scales. Lines and dots indicate continuous spectra as in Fig. 2. (c)–(f) Radial structures of poloidal harmonics of $|\delta u_r|(r|{\omega }_0)$ and $|\delta u_{\zeta }|(r|{\omega }_0)$ for mode M2 of (a) and (b). These are obtained by analyzing slices of data at ${\omega }_0=0.186$ (c),(e) and ${\omega }_0=0.257$ (d),(f).

Figure 4

Global structure and evolution of mode M2 found in the $n=3$ MHD response for ${\beta }_0=3.6\%$. The top panel shows the evolution of the peak amplitudes during one wave period $T_0=2\pi /{\omega }_0=22.4$ for the pressure $\delta P$ (P), and the velocity components $\delta u_{\zeta }$ (S: sound), and $\delta u_r$ (A: Alfvénic) that oscillate with frequency ${\omega }_0=\omega (\mathrm{M}2)=0.257$. Labeled (a)–(e), five snapshots of $\delta P(R,Z|{\omega }_0)$, $\delta u_{\zeta }(R,Z|{\omega }_0)$, and $\delta u_r(R,Z|{\omega }_0)$ are shown as color contour plots. The black ellipses indicate magnetic flux surfaces at $r=0.2$, 0.4, and 0.6. Arrows indicate the (real or apparent) propagation of the structures.

Figure 5

Comparison of NB-driven $n=3$ modes for ${\beta }_0=1.7\%$ and 3.6%. Arranged like Fig. 3. (a),(b) Contour plots of spectrograms $|\delta u_r|(r,\omega )$ computed with $t_{\text{win}}=50$. (c)–(f) Radial structures of poloidal harmonics of $|\delta u_r|(r|{\omega }_0)$ and $|\delta u_{\zeta }|(r|{\omega }_0)$ for frequencies ${\omega }_0$ inferred from (a) and (b). Harmonics of $\delta u_{\zeta }$ belonging to BACMs are shaded in (e),(f).