Nonlinear excitation of energetic particle-induced geodesic acoustic mode via resonance overlap in CFQS

Abstract Energetic particle (EP) driven instabilities are investigated in the Chinese First Quasi-axisymmetric Stellarator using the hybrid kinetic-magnetohydrodynamic code MEGA. EPs are described as the anisotropic slowing-down distribution with pitch angle variable peak value Λ 0 . Simulations reveal that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>0.25</mml:mn> </mml:mrow> </mml:math> or 0.65, two energetic-particle-modes (EPMs) with close growth rates coexist during the linear phase, and poloidal/toroidal mode numbers are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , respectively. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0.3</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mtext>⩽</mml:mtext> <mml:mn>0.6</mml:mn> </mml:mrow> </mml:math> , the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> EPM is the dominant linear instability, and in the nonlinear phase, an EP-induced geodesic acoustic mode (EGAM) is excited and ultimately becomes the dominant instability. The nonlinear excitation mechanism of the EGAM is identified as resonance overlap, based on the analysis of the EP perturbed distribution function δ f . During the linear growth of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> EPM, hole-clump structure of δ f forms within its resonance region in energy E and pitch angle variable Λ space. As the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> EPM saturates and its frequency chirps, the dominant clump structure in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> space migrates towards and gradually overlaps with the EGAM resonance region. This resonance overlap triggers the nonlinear excitation of the EGAM. This is the first demonstration of resonance overlap inducing mode nonlinear excitation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> space and in a stellarator.