Energy scaling of energetic particle transport by microturbulence

Abstract Energetic particle (EP) transport induced by the microturbulence are studied both theoretically and numerically. Based on the quasi-linear theory, the expressions of the four diffusion coefficients describing the EP transport in radial-velocity space ( r , v ) are derived: the radial diffusion coefficient D rr , velocity diffusion coefficient D vv , and two cross-terms D rv and D vr . It is found that D rr scales as E −2 for the deeply trapped particles (TPs), as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for the normally passing particles (PPs), and as E −1 for the purely PPs, where E is the EP energy. The normalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>D</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>D</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>v</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> scale, respectively, as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and E −2 for the purely PPs, and both as E −2 for the deeply TPs. To verify the analytical results, the EP transport induced by the ion temperature gradient turbulence is simulated using GTC. The energy scaling of the simulated diffusion coefficients agrees well with the theoretical predictions. To reveal the underlying physical mechanisms, the finite Larmor radius (FLR) effects, finite orbit width (FOW) effects, and wave-particle resonance condition contributions are investigated separately. The results show that the FLR effects contribute an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> scaling for the TPs and normally PPs, FOW effects contribute an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> scaling for both the deeply TPs and PPs, and resonance condition contributes, respectively, an E −1 scaling for the TPs and an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> scaling for the PPs, consistent with the theoretical analyses. Apart from the energy dependence, the dependence of D rr on the EP pitch is also studied. With the increase of the pitch, the energy scaling is found to exhibit a continuous variation, providing a possible explanation for the discrepancy in previous works on the EP transport energy scaling. Finally, a comparison of the diffusion coefficie