Effects of Plasma Diamagnetic Drift on Alfvén Continua and Discrete Eigenmodes in Tokamaks

Abstract

Alfvén continua and Alfvén eigenmodes (AEs) in DIII-D plasmas (shot #142111) have been studied using a drift-MHD model with self-consistent plasma diamagnetic drift effects. It is shown that the $\omega _*$ causes the frequency upshift in ion diamagnetic drift direction for both Alfvén continua and AEs and thus breaks the ion-electron symmetry of the ideal-MHD model results. The BAE gaps are more sensitive to the $\omega _*$ modification compared to the TAE gaps located in a higher frequency range. After comparing with self-consistent results, an approximate solution of Alfvén continua with $\omega _*$ effect is given with high accuracy. The eigenmode structure of TAE is less influenced by $\omega _*$ effect compared to the real frequency. The incorporation of the diamagnetic drift effects of the background plasmas and energetic particles (EPs) is important to validate theoretical and simulation models against experiments, i.e., correctly address the continuum and radiative dampings as well as AE resonance condition with EPs.

Derivation of Eigenmode Eq. (3) in Magnetic Coordinate

Assuming the eigenvector U in $\left( \psi ,\theta ,\zeta \right)$ coordinates has the form of $U=\sum _{m} U_m\left( \psi \right) exp\left( in\zeta -im\theta \right)$, the terms {I}–{IV} of Eq. (3) in magnetic coordinate are shown by Eqs. (6)–(9) in the following:

$$\begin{aligned} \begin{aligned}&\nabla \cdot \left( \frac{\omega ^2-\omega \omega _*}{V_A^2}\nabla _\perp U\right) \\&\quad = \frac{{\hat{\omega }}^2-{\hat{\omega }}{\hat{\omega }}_*}{J}\left[ \frac{\rho _M}{B_0^2}Jg^{\psi \psi }\frac{\partial ^2U}{\partial \psi ^2} +2\frac{\rho _MJg^{\psi \theta }}{B_0^2}\frac{\partial ^2U}{\partial \psi \partial \theta } +\frac{\rho _M}{B_0^2}Jg^{\theta \theta }\frac{\partial ^2U}{\partial \theta ^2} \right] \\&\qquad +\frac{{\hat{\omega }}^2-{\hat{\omega }}{\hat{\omega }}_*}{J} \left[ \frac{\partial }{\partial \psi }\left( \frac{\rho _M}{B_0^2}Jg^{\psi \psi }\right) +\frac{\partial }{\partial \theta }\left( \frac{\rho _M}{B_0^2}Jg^{\psi \theta }\right) \right] \frac{\partial U}{\partial \psi }\\&\qquad +\frac{{\hat{\omega }}^2-{\hat{\omega }}{\hat{\omega }}_*}{J} \left[ \frac{\partial }{\partial \psi }\left( \frac{\rho _M}{B_0^2}Jg^{\psi \theta }\right) +\frac{\partial }{\partial \theta }\left( \frac{\rho _M}{B_0^2}Jg^{\theta \theta }\right) \right] \frac{\partial U}{\partial \theta }, \end{aligned} \end{aligned}$$

(6)

where ${\hat{\omega }}=\omega /\left( V_{ap}/R_0\right)$, ${\hat{\omega }}_*=\omega _*/\left( V_{ap}/R_0\right)$, $V_{Ap} = B_a/\sqrt{4\pi n_am_p}$ is the proton Alfvén speed at magnetic axis, $B_a$ and $n_a$ are the magnetic field and plasma number density at magnetic axis, $m_p$ is the proton mass, $\rho _M=n_0m_i$ is the plasma mass density, $g^{\psi \psi }=\nabla \psi \cdot \nabla \psi$, $g^{\theta \theta }=\nabla \theta \cdot \nabla \theta$, and $g^{\psi \theta }=\nabla \psi \cdot \nabla \theta$.

$$\begin{aligned} \begin{aligned}&{\mathbf {B}}_{\mathbf {0}}\cdot \nabla \left[ \frac{1}{B_0^2}\nabla \cdot \left( B_0^2\nabla _\perp Q\right) \right] \\&\quad = {\mathbf {B}}_{\mathbf {0}}\cdot \nabla \left\{ g^{\psi \psi }\frac{\partial ^2Q}{\partial \psi ^2} +2g^{\psi \theta }\frac{\partial ^2Q}{\partial \psi \partial \theta } +g^{\theta \theta }\frac{\partial ^2Q}{\partial \theta ^2}\right. \\&\quad \left. +\frac{1}{JB_0^2}\left[ \frac{\partial }{\partial \psi }\left( JB_0^2g^{\psi \psi }\right) +\frac{\partial }{\partial \theta }\left( JB_0^2g^{\psi \theta }\right) \right] \frac{\partial Q}{\partial \psi }\right. \\&\quad \left. +\frac{1}{JB_0^2}\left[ \frac{\partial }{\partial \psi }\left( JB_0^2g^{\psi \theta }\right) +\frac{\partial }{\partial \theta }\left( JB_0^2g^{\theta \theta }\right) \right] \frac{\partial Q}{\partial \theta } \right\} , \end{aligned} \end{aligned}$$

(7)

where ${\mathbf {B}}_{\mathbf {0}}\cdot \nabla =i\frac{q}{J}{\widetilde{k}}_m$, $Q=\frac{1}{B_0^2}{\mathbf {B}}_{\mathbf {0}}\cdot \nabla U =i\frac{q}{JB_0^2}{\widetilde{k}}_mU_m$, and ${\widetilde{k}}_m=n-m/q$.

$$\begin{aligned}&-\frac{4\pi }{c}\left( \nabla Q\times {\mathbf {B}}_{\mathbf {0}}\right) \cdot \nabla \left( \frac{J_{||}}{B_0}\right) \nonumber \\&\quad =-\frac{1}{J}\left( g\frac{\partial Q}{\partial \theta }-I\frac{\partial Q}{\partial \zeta }\right) \frac{\partial }{\partial \psi }\left[ \frac{1}{JB_0^2}\left( -I\frac{\partial g}{\partial \psi }+g\frac{\partial I}{\partial \psi }\right) \right] . \end{aligned}$$

(8)

$$\begin{aligned}&-\frac{8\pi }{B_0^2}\varvec{\kappa }\cdot \nabla \delta P\times {\mathbf {B}}_{\mathbf {0}} =-\frac{8\pi }{JB_0^2}\left[ \kappa _\psi \left( g\frac{\partial \delta P}{\partial \theta }- I \frac{\partial \delta P}{\partial \zeta }\right) -\kappa _\theta g \frac{\partial \delta P}{\partial \psi }\right] , \end{aligned}$$

(9)

where the magnetic curvature $\varvec{\kappa }=\left( \nabla \times {\mathbf {b}}_{\mathbf {0}}\right) \times {\mathbf {b}}_{\mathbf {0}}=\kappa _\psi \nabla \psi + \kappa _\theta \nabla \theta + \kappa _\zeta \nabla \zeta$, $\kappa _\psi =\frac{1}{B_0}\frac{\partial B_0}{\partial \psi }-\frac{1}{JB_0^2}\left( \frac{\partial I}{\partial \psi }+q\frac{\partial g}{\partial \psi }\right)$, $\kappa _\theta =\frac{gq}{JB_0^3}\frac{\partial B_0}{\partial \theta }$, and $\kappa _\zeta =-\frac{g}{JB_0^3}\frac{\partial B_0}{\partial \theta }$. The perturbed pressure in Eq. (9) can be written as $\delta P =\frac{i}{JB_0^2}\left( mg+nI\right) \frac{\partial P_0}{\partial \psi }U -\frac{2\Gamma P_{mhd}}{JB_0^2}\left[ i\kappa _\psi \left( mg+nI\right) +\kappa _\theta g\frac{\partial }{\partial \psi }\right] U$, $P_0=P_{mhd}+P_{f}$ and $\Gamma P_{mhd} = P_e + \left( 7/4\right) P_i$.