On electromagnetic perturbations of geodesic acoustic modes in anisotropic tokamak plasmas

Abstract Electromagnetic geodesic acoustic modes are analytically investigated in tokamak plasmas with anisotropy, utilizing gyro-kinetic equations and a rigorously self-consistent anisotropic distribution. When including first-order finite-orbit-width effects and first-order finite-Larmor-radius effects, it is proven that the anisotropy with an arbitrary strength does not induce the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> harmonics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:msub> </mml:mrow> </mml:math> , where m and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:msub> </mml:mrow> </mml:math> denote the poloidal wavenumber and the parallel component of the perturbed magnetic vector potential, respectively. The rigorously self-consistent anisotropy introduces an equilibrium electrostatic field with poloidally asymmetric structure, and consequently induces an additional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>E</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow> <mml:mover> <mml:mi>B</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> drift term within the gyro-kinetic equation. This equilibrium electrostatic field inhibits the anisotropy from generating non-zero <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> harmonics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:msub> </mml:mrow> </mml:math> . Indeed we demonstrate that introducing anisotropy self-consistently into the equilibrium quantitatively influences <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> harmonics of the perturbed electrostatic potential, but only the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> harmonics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:msub> </mml:mrow> </mml:math> .