Physics of Alfvén waves and energetic particles in burning plasmas

Dynamics of shear Alfvén waves and energetic particles are crucial to the performance of burning fusion plasmas. This article reviews linear as well as nonlinear physics of shear Alfvén waves and their self-consistent interaction with energetic particles in tokamak fusion devices. More specifically, the review on the linear physics deals with wave spectral properties and collective excitations by energetic particles via wave-particle resonances. The nonlinear physics deals with nonlinear wave-wave interactions as well as nonlinear wave-energetic particle interactions. Both linear as well as nonlinear physics demonstrate the qualitatively important roles played by realistic equilibrium nonuniformities, magnetic field geometries, and the specific radial mode structures in determining the instability evolution, saturation, and, ultimately, energetic-particle transport. These topics are presented within a single unified theoretical framework, where experimental observations and numerical simulation results are referred to elucidate concepts and physics processes.

Figure 1

(a) The generation of the distribution $\delta f_k$ due to the interaction of $f_0$ with the field $\delta {\varphi }_k$, corresponding to the solution of Eq. (4.126). (b) Nonlinear distortion of $f_0$ due to emission and absorption of the field $\delta {\varphi }_k$. (c) The process is defined in the top frame, while the solution of the Dyson equation (4.128) corresponds to the summation of all terms in the Dyson series (bottom) ([7]).

Figure 2

(a)  TAE saturation amplitude vs the normalized linear growth rate, expressed in Alfvén time units ${\tau }_A=R_0/v_A$ computed at the magnetic axis and with $R_0$ denoting the geometric center of the circular toroidal plasma ([86]). From [89]. (b) BAE saturation amplitude, expressed by the peak scalar potential energy normalized with respect to the EP birth energy, whose distribution function is an isotropic slowing down, is shown vs the normalized mode linear growth rate. From [496].

Figure 3

Radial profiles of ${\beta }_E$ and $(m,n=2)$ Fourier components of the EPM scalar potential fluctuations during the linear growth (left), the end of the EPM avalanche (middle), and saturation phase (right). Time normalization is ${\omega }_{A}t$, with ${\omega }_A=v_A/R_0$ computed at the magnetic axis. From [481].

Figure 4

Radial position $(r/a{)}_{\max }$ (top) and value of the maximum gradient $[d(r{\beta }_E)/dr{]}_{\max }$ vs ${\omega }_{A}t$ for the EPM simulation in Fig. 3. The strong convection, characteristic of the avalanche phase, is accompanied by gradient steepening, followed by a relaxation phase, characterized by diffusive EP transport. From [481].

Figure 5

The functions $W(x)$ and $\phi (x)$ describing the self-similar shape $U(x)=W(x)e^{i\phi (x)}$ of the EPM wave-packet propagation in the early phase of its nonlinear evolution ([544], [558]).

Figure 6

(a) Evolution of the fishbone frequency vs time. The frequency is expressed in units of ${\omega }_{A0}=v_{A0}/R_0$ and the time in units of ${\omega }_{A0}^{-1}$. (b) Evolution of the resonant EP distribution function for $v/v_{A0}=0.8$ and $\mu B_0/\mathcal{E}=1$. From [203].

Figure 7

(a) Time evolution of ${\overline{\omega }}_D$ (red line) and ${\overline{\omega }}_D\pm \delta {\omega }_D$ (dashed red lines), compared with the time evolving mode frequency from simulation results (black line). (b) Time evolution of $\overline{\dot{\mathrm{\Theta }}}$ (black line) and $\overline{\dot{\mathrm{\Theta }}}\pm \delta \dot{\mathrm{\Theta }}$ (dashed black lines); $\overline{\dot{\mathrm{\Theta }}}{|}_{\omega =\text{const}}$, obtained neglecting frequency chirping is also shown (red line). From [484]. (c) Time evolution of $\overline{r}$ (middle, black line) and of $\overline{r}\pm \delta r$ (top and bottom, red lines). The linear mode structure is also shown by $|(m/r)\delta {\varphi }_{m,n}|\propto |\delta {\mathit{\xi }}_{rm,n}|$ in abscissa vs the normalized radial position on the vertical axis. The harmonic in red refers to the dominant $(m,n)=(1,1)$ component. From [484].