Nonlinear excitation of energetic particle driven geodesic acoustic mode by resonance overlap with Alfvén instability in ASDEX Upgrade

Abstract

The Alfvén instability nonlinearly excited the energetic-particle-driven geodesic acoustic mode on the ASDEX-Upgrade tokamak, as demonstrated experimentally. The mechanism of the energetic-particle-driven geodesic acoustic mode excitation and the mode nonlinear evolution is not yet fully understood. In the present work, a first-principles simulation using the MEGA code investigated the mode properties in both the linear growth and nonlinear saturated phases. Here we show that the simulation successfully reproduced the excitation and coexistence of these two modes, and agreed with the experimental results well. Conclusive evidence showed that the resonance overlap is the excitation mechanism of the energetic-particle-driven geodesic acoustic mode. In the linear growth phase, energetic particles that satisfied different resonance conditions excited the Alfvén instability, which then caused energetic particle redistribution in phase space. These redistributed energetic particles caused resonance overlap, exciting the energetic-particle-driven geodesic acoustic mode in the nonlinear phase.

Fig. 1

The reproduction of experimental phenomena by simulation. The frequency spectrum of the Alfvén instability and EGAM are shown, where the Alfvén instability is the higher frequency mode and the EGAM is the lower frequency mode. Panel (a) shows the frequency spectrum in the simulation, the color bar represents the velocity perturbation normalized by Alfvén velocity and is plotted on a logarithmic scale. Panel (b) shows the experimental observation in shot #34924 of AUG. The color bar represents the soft X-ray emission power and is plotted on a logarithmic scale. Yellow dashed line and pink dotted line represent the excitation time of Alfvén instability and EGAM.

Fig. 2

The resonance overlap illustrated by the energetic particle distribution $f_{\mathit{total}}$ in phase space. The $f_{\mathit{total}}$ in ($P_{\varphi },E$) phase space for different $\mu$ values at different times are shown. $P_{\varphi }$ is normalized to the product of particle charge $e_{\mathit{EP}}$ and the maximum $\psi$ at the plasma center. From the top to the bottom, the three rows are plotted at $t=0.304\text{ms}$, $0.375\text{ms}$, and $0.609\text{ms}$, respectively. From the left to the right, the five columns represent different $\mu$ values of 1.62, 3.56, 5.51, 7.45 and 9.39 with unit keV/T, respectively. The black color represents the minimum $f_{\mathit{total}}$ value 0. The bright yellow color represents the maximum $f_{\mathit{total}}$ values, and in the five columns from the left to the right, they are 30, 12, 5, 4, and 2.5, respectively. The solid and dashed white curves represent respectively $L_{\mathit{Alv}}=1$ and 0, and the green curve represents $L_{\mathit{EGAM}}=1$. The two cyan dotted lines represent two constant $E^{\prime }$ values. The constant $P_{\varphi }$ lines are not plotted because they are parallel to the vertical axis.

Fig. 3

The resonance overlap illustrated by the energetic particle distribution $f_{\mathit{total}}$ with detailed time evolution. The time evolution of $f_{\mathit{total}}$ along (a) the left (or higher) $E^{\prime }$ line of Fig. 2 and (b) the vertical line of Fig. 2 with a $P_{\varphi }/e_{\mathit{EP}}{\psi }_{\mathit{max}}$ value of 0.778, are shown in details. The particle $\mu$ value is $7.45\mathrm{keV}/\mathrm{T}$. These three horizontal lines from top to bottom represent three resonant layers of $L_{\mathit{Alv}}=1$, $L_{\mathit{EGAM}}=1$, and $L_{\mathit{Alv}}=0$, respectively. To better understand the physical pictures described above, the bird’s-eye view 3-dimensional sub-figures are also presented. The sub-figures (c) and (d) correspond to the sub-figures (a) and (b), respectively, and their vertical axes represent $f_{\mathit{total}}$. The sub-figures (e–g) show the $f_{\mathit{total}}$ along $P_{\varphi }=const.$ at different times, the $P_{\varphi }/e_{\mathit{EP}}{\psi }_{\mathit{max}}$ values are 0.994, 0.778, and 0.804, the maximum values of the vertical axis are 6.5, 3.5 and 3.5, and the particle $\mu$ values are 5.51, 7.45 and $9.39\mathrm{keV}/\mathrm{T}$, respectively. The vertical dotted lines represent the resonant layers of $L_{\mathit{EGAM}}=1$.

Fig. 4

The resonance overlap illustrated by the energetic particle distribution $\delta f$ in phase space. The $\delta f$ in ($P_{\varphi },E$) phase space for different $\mu$ values at different times are shown. $P_{\varphi }$ is normalized to the product of particle charge $e_{\mathit{EP}}$ and the maximum $\psi$ at the plasma center. From the top to the bottom, the three rows are plotted at $t=0.375\text{ms}$, $0.516\text{ms}$ and $0.656\text{ms}$, respectively. From the left to the right, the five columns represent different $\mu$ values of 1.62, 3.56, 5.51, 7.45 and 9.39 with unit keV/T, respectively. The red color represents positive $\delta f$ and the blue represents negative $\delta f$. The solid and dashed black curves represent respectively $L_{\mathit{Alv}}=1$ and 0, and the red curve represents $L_{\mathit{EGAM}}=1$. The two black dotted lines represent two constant $E^{\prime }$ values. The constant $P_{\varphi }$ lines are not plotted because they are parallel to the vertical axis.