A solution method of slowing-down distribution of energetic particles in tokamaks

Dai, Yongzhi; Cao, Jinjia; Xiang, Dong; Yang, Junhui

doi:10.1063/5.0123241

Physics of Plasmas

2023

DOI: 10.1063/5.0123241

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A solution method of slowing-down distribution of energetic particles in tokamaks

Yongzhi Dai

Jinjia Cao

Dong Xiang

et al.

Abstract: The eigen equation of pitch-angle distribution derived from the slowing-down distribution equation with an energetic particle source term is solved by using the Legendre series expansion method. An iteration matrix is established when pitch-angle scattering terms become important. The whole pitch-angle region is separated into three parts, two passing regions, and one trapped area. The slowing-down distribution for each region is finally obtained. The method is applied to solve the slowing-down equations with … Show more

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Cited by 4 publications

References 27 publications

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Simulation of neutral beam current drive on EAST tokamak

Hu,

Xu,

et al. 2023

Physics of Plasmas

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A neutral beam current drive on the EAST tokamak is studied by using Monte Carlo test particle code TGCO. The phase-space structure of the steady-state fast ion distribution is examined and visualized. We find that trapped ions carry co-current current near the edge and countercurrent current near the core. However, the magnitude of the trapped ion current is one order smaller than that of the passing ions. Therefore, their contribution to the fast ion current is negligible (1% of the fast ion current). We examine the dependence of the fast ion current on two basic plasma parameters: the plasma current Ip and plasma density ne. The results indicate that the dependence of fast ion current on Ip is not monotonic: with Ip increasing, the fast ion current first increases and then decreases. This dependence can be explained by the change of trapped fraction and drift-orbit width with Ip. The fast ion current decreases with the increase in plasma density ne. This dependence is related to the variation of the slowing-down time with ne, which is already well known and is confirmed in our specific situation. The electron shielding effect to the fast ion current is taken into account by using a fitting formula applicable to general tokamak equilibria and arbitrary collisionality regime. The dependence of the net current on the plasma current and density follows the same trend as that of the fast ion current.

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Simulation of neutral beam current drive on EAST tokamak

Hu,

Xu,

et al. 2023

Physics of Plasmas

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Alpha particle distribution for full energy region

Zhang,

Wang,

Dong

et al. 2024

Physics of Plasmas

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The alpha particle distribution in the full energy region is studied in this work. The Fokker–Planck equation near the equilibrium state is analytically solved using an iteration method for alpha particles. The resulting distribution includes the Maxwellian part, the modified slowing down part, and the high energy tail part. The analytical results align with the conventional slowing down distribution in the high energy region vα>0.2vb and the Maxwellian distribution in the low energy region vα<0.1vb, where vα is the alpha particle velocity and vb is the birth velocity of alpha particles. In the range of 0.1vb<vα<0.2vb, the distribution is described by the sum of the Maxwellian part and modified slowing down part. Turbulent transport effects on the alpha particle distribution are studied through introducing a local sink term. A simplified analytical solution is provided, and the bump-on-tail distribution is observed under appropriate parameters. In both cases with and without transport effects, the Fokker–Planck equation is numerically solved. It is shown that the full temporal evolution of the system can be divided into three stages: the slowing down stage, thermalization stage, and quasi-steady state stage. The numerically obtained alpha particle distribution in the quasi-steady state stage agrees well with the theoretical results.

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A numerical method for calculating the driven current of neutral beam injection in tokamaks

Chen,

Cao,

Dai

et al. 2024

AIP Advances

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The current driven by neutral beam injection in tokamak is calculated, and the slowing-down distribution function of the fast ion is obtained by the backward Euler iteration method, including the pitch angle scattering collision. This study reveals that when the pitch-angle cosine is small, the trapped fast-ion current significantly contributes to the total driven current, particularly when the neutral beam is injected perpendicularly. In such cases, the current densities of passing and trapped ions are of the same order of magnitude, with the trapped fast-ion current contributing over 10% to the total neutral beam-driven current. This results in a parabolic profile of the total current in the radial direction, promoting the formation of a negative shear equilibrium structure in the core of the tokamak plasma. The numerical approach was validated against the NUBEAM code while considering electron shielding effects and applied to calculate the neutral beam-driven current in multiple tokamaks. The influence of pitch-angle cosine and neutral beam injection power on the driven current was studied at different radial positions.

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A solution method of slowing-down distribution of energetic particles in tokamaks

Cited by 4 publications

References 27 publications

Simulation of neutral beam current drive on EAST tokamak

Simulation of neutral beam current drive on EAST tokamak

Alpha particle distribution for full energy region

A numerical method for calculating the driven current of neutral beam injection in tokamaks

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