Speaker
Description
We have developed a two-fluid model of the energetic-particle-driven geodesic acoustic mode (EGAM) in magnetically confined toroidal plasmas by applying fluid models to the bulk plasma and energetic particles separately. Analysis of the theoretical model reveals two EGAM solutions with high and low frequencies. In the high-frequency mode, the pressure perturbations of the two components oscillate in phase, whereas they oscillate out of phase in the low-frequency mode. In this presentation, the analytical results are compared with gyrokinetic theory [1] and simulation results [2].
We consider an electrostatic magnetohydrodynamic (MHD) oscillation in a toroidal plasma with concentric circular magnetic surfaces. Let $r$ denote the minor radius, $q(r)$ the safety-factor profile, and $R_0$ the major radius at the plasma center. The bulk plasma and energetic particles are denoted by the subscripts $b$ and $h$, respectively. The equilibrium density and pressure profiles are denoted by $\rho_{s0}(r)$ and $P_{s0}(r)$ ($s=b$ or $h$), and a static equilibrium with no plasma flow is considered.
By coupling the linear equations for the radial electric field, the pressure perturbations of the two components, and their velocity perturbations parallel to the magnetic field, we analyze an axisymmetric mode (toroidal mode number $n=0$) under the large-aspect-ratio approximation ($R_0 \gg r$). When the poloidal mode number of the radial electric field is $m=0$, the pressure and parallel-velocity perturbations depend on $\sin \theta$ and $\cos \theta$, respectively. The analysis yields the following solutions for the eigenfrequency $\omega$.
\begin{equation}
\omega^2 = \frac{1}{2}\left( \omega_{Gb}^2 + \omega_{Gh}^2
\pm \sqrt{( \omega_{Gb}^2 - \omega_{Gh}^2)^2 + 4\Omega_b^2 \Omega_h^2} \right) \ .
\end{equation}
Here, $\omega_s^2 \equiv \gamma_s P_{s0}/(\rho_{s0} q^2 R_0^2)$, $\Omega_s^2 \equiv 2 \gamma_s P_{s0}/(\rho_{0}R_0^2)$, and $\omega_{Gs}^2\equiv \omega_s^2 + \Omega_s^2$.
Note that $\Omega_s$ is defined using the total density $\rho_0 \equiv \rho_{b0}+\rho_{h0}$ rather than the density $\rho_s$ of each component. Here, $\gamma_s$ is the ratio of specific heats. In the limit where the energetic-particle density, pressure, and $\Omega_h$ are negligibly small ($\rho_{h0}\simeq 0$, $P_{h0}\simeq 0$, $\Omega_{h}\simeq 0$), the two solutions converge to
\begin{equation}
\omega = \omega_\mathrm{GAM}\equiv \frac{1}{R_0} \sqrt{ \frac{\gamma_b P_{b0}}{\rho_{b0}} \left(2+\frac{1}{q^2} \right)} \ , \qquad
\omega = \omega_h \ .
\end{equation}
The first solution is the geodesic acoustic mode (GAM) in the absence of energetic particles. The second solution is the frequency of the energetic-particle acoustic wave, which arises from compression associated with the parallel flow velocity.
[1] G.Y. Fu, Phys. Rev. Lett. 101, 185002 (2008).
[2] Hao Wang et al., Phys. Rev. Lett. 120, 175001 (2018).