Stabilization of ion-temperature-gradient mode by trapped fast ions

Siqi WANG; Huishan CAI; Baofeng GAO; Ding LI

doi:10.1088/2058-6272/ac5e73

Plasma Science and Technology > 2022 > 24(6): 065102. > DOI: 10.1088/2058-6272/ac5e73

Siqi WANG, Huishan CAI, Baofeng GAO, Ding LI. Stabilization of ion-temperature-gradient mode by trapped fast ions[J]. Plasma Science and Technology, 2022, 24(6): 065102. DOI: 10.1088/2058-6272/ac5e73

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Stabilization of ion-temperature-gradient mode by trapped fast ions

1.
CAS Key Laboratory of Geospace Environment, School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, People's Republic of China
2.
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China

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Author Bio:
Huishan CAI, E-mail: hscai@mail.ustc.edu.cn
Received Date: October 21, 2021
Revised Date: March 13, 2022
Accepted Date: March 15, 2022
Available Online: December 12, 2023
Published Date: May 17, 2022

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Abstract

Abstract

Understanding and modeling fast-ion stabilization of ion-temperature-gradient (ITG) driven microturbulence have profound implications for designing and optimizing future fusion reactors. In this work, an analytic model is presented, which describes the effect of fast ions on ITG mode. This model is derived from a bounce-average gyro-kinetic equation for trapped fast ions and ballooning transformation for ITG mode. In addition to dilution, strong wave-fast-ion resonant interaction is involved in this model. Based on numerical calculations, the effects of the main physical parameters are studied. The increasing density of fast ions will strengthen the effects of fast ions. The effect of wave-particle resonance strongly depends on the temperature of fast ions. Furthermore, both increasing density gradient and the ratio of the temperature and density gradients can strengthen the stabilization of fast ions in ITG mode. Finally, the influence of resonance broadening of wave-particle interaction is discussed.
- fast ions,
- ITG instability,
- gyro-kinetic

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1. Introduction

Improving plasma confinement is beneficial for designing future nuclear fusion devices and optimizing the performance of present devices. An important limiting factor of plasma confinement in fusion devices is microturbulence [1]. As a significant driver of plasma turbulence, ion-temperature-gradient (ITG) instability [2–5] is principally responsible for the degradation of ion energy confinement. Therefore, it is extremely valuable to study the mechanisms that limit its development.

Fast ions [6] are mainly generated by fusion reactions, as well as auxiliary heating systems, such as neutral beam injection [7] and ion cyclotron resonance heating [8]. Understanding the behavior of fast ions is important since they are an essential component of fusion plasma and play a major role in sustaining fusion-relevant bulk temperatures. In addition, fast ions carry large power, which implies that even small fast-ion losses can damage the first wall of a fusion device. Therefore, the confinement of fast ions is worth studying.

The study of fast-ion interaction with plasma turbulence has recently attracted particular interest. On the one hand, background plasma turbulence can induce the transport of fast ions and lead to the redistribution or losses of fast ions [9–12]. On the other hand, fast ions can affect plasma turbulence in turn [13–15]. In some experimental and numerical studies [16–20], the suppression of plasma turbulence has also been observed, which is linked to the presence of fast ions. Generally, the effects of fast ions can be classified as electrostatic effects and electromagnetic effects. In [14], the effect of the dilution of the main ions on stabilizing ITG turbulence is investigated. However, the fast-ion effect can be observed even under low density, which suggests other interaction mechanisms in addition to dilution [21]. In [22], a strong dynamic effect of fast ions on suppressing plasma turbulence has been observed in an electrostatic setup. Significantly, the wave-particle resonance mechanism is taken into account [23, 24]. With regard to electromagnetic stabilization of ITG turbulence, there are many experimental and numerical studies [25–30]. As shown in [22], the linear growth rate exhibits the same behavior in the electromagnetic and electrostatic framework, while the growth rate is lower in the electromagnetic framework than that in the electrostatic framework for the same parameters. Mostly, the effects of the fast-ion stabilization of ITG turbulence are studied by both linear and nonlinear numerical simulation methods.

In this paper, a theoretical interpretation for the observed impact of fast ions on ITG mode is offered. Both analytical and numerical calculations are presented in this work. The physical mechanisms of fast-ion stabilization on ITG mode are studied in detail. The interaction of fast ions with ITG mode is investigated in the framework of gyro-kinetic theory [31]. The dispersion relation consisting of ITG mode and fast ions is normally derived by utilizing the quasi-neutrality condition and ballooning transformation. It is discovered that fast ions can interact with ITG instability via wave-particle resonance. Based on numerical calculations, the effects of density, temperature, density gradient and the ratio of the temperature and density gradients of fast ions on both the real frequency and growth rate of ITG mode are investigated. The effect of dilution is also presented in numerical calculations. It is found that, in addition to dilution, the resonant fast-ion stabilizing effect plays a significant role.

The rest of the paper is organized as follows. In section 2, adopting proper approximations, the dispersion relation including ITG mode and trapped fast ions is established. In section 3, based on numerical calculations, the dispersion relation is solved and different physical parameters controlling the stabilizing effect are investigated. Furthermore, the influence of resonance broadening of wave-particle interaction is discussed. Our conclusions are given in section 4.

2. Dispersion relation

The dispersion relation for ITG mode including fast ions and the bulk plasma is given by quasi-neutrality equation:

$δ n_{e} = δ n_{i} + Z_{f} δ n_{f} .$

(1)

Here, the bulk plasma is perceived to be composed of deuterium and electrons. Z_f is the charge number of fast ions. For the electron response, an adiabatic approximation is adopted for k_‖v_{T_e} ≫ ω and the electron perturbation density δn_e is given by,

$δ n_{e} = n_{e 0} \frac{|e| δ ϕ}{T_{e}} .$

(2)

However, an adiabatic approximation is not appropriate for both main ions and fast ions. The perturbed ion distribution function can be written as,

$δ f_{j} = - \frac{Z_{j} |e| δ ϕ}{T_{j}} F_{Mj} + δ g_{j} .$

(3)

Here, j refers to the species of particles and δg_j is the non-adiabatic part of the perturbed distribution function, which is determined by the gyro-kinetic equation [31]:

$\begin{matrix} (\frac{\partial}{\partial t} + v_{‖} \nabla_{‖} + v_{d j} \cdot \nabla_{⊥}) δ g_{j} \\ - i \frac{Z_{j} e δ ϕ}{M_{j}} J_{0} [\frac{k_{θ}}{ω_{c j}} \frac{\partial F_{Mj}}{\partial r} + ω \frac{\partial F_{Mj}}{\partial K}] = 0 . \end{matrix}$

(4)

Here, $v_{d j} = (v_{‖}^{2} + v_{⊥}^{2} / 2) b \times κ / ω_{c j}$ , ω_cj = Z_jeB/M_j and K = v²/2, where b is a unit vector parallel to the magnetic field, κ is the magnetic curvature, M_j is the mass of species j and $(r, θ, ξ)$ denote the minor radius, poloidal angle and toroidal angle, respectively. The equilibrium distribution function is assumed to be Maxwellian. Obviously, in the above equation, the first term in the square bracket denotes the space-gradient-driven term and the second term denotes the energy-gradient-driven term.

First, the dynamics of background main ions is presented. According to the gyro-kinetic equation, the perturbed ion distribution function δg_i is obtained as,

$δ g_{i} = \frac{ω - ω_{* i} [1 + η_{i} (\frac{E}{T_{i}} - \frac{3}{2})]}{ω - k_{‖} v_{‖} - ω_{di}} \frac{|e| δ ϕ}{T_{i}} J_{0} F_{Mi},$

(5)

where,

$\begin{matrix} k_{‖} & = & - i \frac{1}{qR} (\frac{\partial}{\partial θ} + i {sk}_{θ} x), \\ ω_{di} & = & {\bar{ω}}_{di} (\frac{v_{‖}^{2}}{v_{T_{i}}^{2}} + \frac{v_{⊥}^{2}}{2 v_{T_{i}}^{2}}), \\ {\bar{ω}}_{di} & = & ϵ_{n} ω_{* i} (\cos θ + i \frac{\sin θ}{k_{θ}} \frac{\partial}{\partial x}), \end{matrix}$

(6)

with $ω_{* j} = k_{θ} {cT}_{j} / |e| {BL}_{n_{j}}$ , $v_{T_{j}} = \sqrt{T_{j} / M_{j}}$ , $ϵ_{n} = L_{n_{e}} / R$ , $L_{n_{j}} = {|\nabla n_{j} / n_{j}|}^{- 1}$ , $L_{T_{j}} = {|\nabla T_{j} / T_{j}|}^{- 1}$ and $η_{j} = L_{n_{j}} / L_{T_{j}}$ . q is the safety factor, $s = {rq}^{'} / q$ is the magnetic shear and the radial variable x is the distance from a reference mode rational surface.

In previous works [3, 4], the above equation has been studied under some limits. By considering the limits $k_{‖} v_{T_{i}} / ω ≪ 1$ , ω_di/ω ≪ 1 and keeping the leading contributions in $k_{‖} v_{T_{i}} / ω$ and ω_di/ω, the perturbed density of the main ions is expressed as,

$\begin{matrix} δ n_{i} = - n_{i 0} \frac{|e| δ ϕ}{T_{i}} + [(1 - \frac{ω_{* i}}{ω}) (1 + 2 \frac{{\bar{ω}}_{di}}{ω} + \frac{k_{‖}^{2} v_{‖}^{2}}{ω^{2}} - b_{i}) \\ - \frac{ω_{* i}}{ω} η_{i} (2 \frac{{\bar{ω}}_{di}}{ω} + \frac{k_{‖}^{2} v_{‖}^{2}}{ω^{2}} - b_{i})] \frac{n_{i 0} |e| δ ϕ}{T_{i}}, \end{matrix}$

(7)

with $b_{i} = k_{⊥}^{2} ρ_{T_{i}}^{2}$ . Here, ρ_{T_i} is the Larmor radius of the background main ions. Note that, under the limits ( $k_{‖} v_{T_{i}} / ω ≪ 1$ , ω_di/ω ≪ 1), the growth rate of ITG mode in this work is larger than that in simulation studies [22].

For the fast-ion dynamics, both of δg_f and δϕ are expanded in toroidal and poloidal Fourier harmonics:

$\begin{matrix} δ g_{f} & = & \sum_{m, n} δ {\hat{g}}_{f}^{m, n} (r, θ) e^{i n ξ - i m θ}, \\ δ ϕ & = & \sum_{m, n} δ {\hat{ϕ}}^{m, n} (r) e^{i n ξ - i m θ} . \end{matrix}$

(8)

Then, the Fourier transform of equation (4) is,

$\begin{matrix} \frac{v_{‖}}{qR} \sum_{m, n} \frac{\partial δ {\hat{g}}_{f}^{m, n}}{\partial θ} - \frac{v_{‖}}{qR} \sum_{m, n} i (m - nq) δ {\hat{g}}_{f}^{m, n} \\ - i ω \sum_{m, n} δ {\hat{g}}_{f}^{m, n} - \frac{{\bar{v}}_{df} \cos θ}{r} \sum_{m, n} \frac{\partial δ {\hat{g}}_{f}^{m, n}}{\partial θ} \\ + i \frac{{\bar{v}}_{df} \cos θ}{r} \sum_{m, n} m δ {\hat{g}}_{f}^{m, n} - i k_{r} {\bar{v}}_{df} \sin θ \sum_{m, n} δ {\hat{g}}_{f}^{m, n} \\ = - i ω \frac{Z_{f} |e|}{T_{f}} J_{0} F_{Mf} \sum_{m, n} δ {\hat{ϕ}}^{m, n} \\ + i [1 + η_{f} (\frac{E}{T_{f}} - \frac{3}{2})] \frac{Z_{f} |e|}{T_{f}} J_{0} F_{Mf} \sum_{m, n} ω_{* f}^{m, n} δ {\hat{ϕ}}^{m, n} . \end{matrix}$

(9)

Here, ${\bar{v}}_{df} = v_{⊥}^{2} / (2 R ω_{cf})$ and $v_{‖}^{2}$ has been neglected for trapped particles. E is kinetic energy. Note that the two terms on the right-hand side of the above equation are obtained from the energy-gradient-driven term and the space-gradient-driven term in equation (4), respectively.

The ordering relationship between the terms in equation (9) is as follows:

$\begin{matrix} 1 : \frac{∆ r}{∆ r_{m}} : \frac{ω}{ω_{tf}} : \frac{ρ_{T_{f}}}{r} : k_{θ} ρ_{T_{f}} : k_{r} ρ_{T_{f}} : \frac{ω}{ω_{tf}} \frac{δ {\hat{f}}_{a}^{m, n}}{δ {\hat{g}}_{f}^{m, n}} : \\ \frac{ω_{* f}^{T}}{ω_{tf}} \frac{δ {\hat{f}}_{a}^{m, n}}{δ {\hat{g}}_{f}^{m, n}} . \end{matrix}$

(10)

Here, ∆r is the radial distance from a reference mode rational surface and ∆r_m is the distance between the two adjacent rational surfaces, ∆r/∆r_m < 1. ω/ω_tf < 1 implies $T_{f} / T_{i} > k_{θ}^{2} ρ_{T_{i}}^{2} R^{2} / L_{n_{i}}^{2}$ , where ω_tf is the transit frequency. It is also true that ρ_{T_f}/r < 1 for ITG mode. k_θρ_{T_f} < 1 requires $T_{f} / T_{i} < 1 / k_{θ}^{2} ρ_{T_{i}}^{2}$ . Note that the upper and lower limits of T_f/T_i are given as $k_{θ}^{2} ρ_{T_{i}}^{2} R^{2} / L_{n_{i}}^{2} < T_{f} / T_{i} < 1 / k_{θ}^{2} ρ_{T_{i}}^{2}$ . Subsequently, k_rρ_{T_f} < 1 for k_r ~ sk_θ. $δ {\hat{f}}_{a}^{m, n} = Z_{f} |e| δ {\hat{ϕ}}^{m, n} F_{Mf} / T_{f}$ denotes the adiabatic part of the perturbed distribution function of fast ions. Then $ω δ {\hat{f}}_{a}^{m, n} / ω_{tf} δ {\hat{g}}_{f}^{m, n} < 1$ and $ω_{* f}^{T} δ {\hat{f}}_{a}^{m, n} / ω_{tf} δ {\hat{g}}_{f}^{m, n} < 1$ , where $ω_{* f}^{T} = ω_{* f} [1 + η_{f} (E / T_{f} - 3 / 2)]$ .

Based on the above ordering, equation (9) can be expanded as,

$\frac{v_{‖}}{qR} \sum_{m, n} \frac{\partial δ {\hat{g}}_{f 0}^{m, n} (r, θ)}{\partial θ} = 0,$

(11)

$\begin{matrix} \sum_{m, n} [\frac{v_{‖}}{qR} \frac{\partial δ {\hat{g}}_{f 1}^{m, n} (r, θ)}{\partial θ} - i \frac{v_{‖}}{qR} (m - nq) δ {\hat{g}}_{f 0}^{m, n} (r, θ) \\ - i ω δ {\hat{g}}_{f 0}^{m, n} (r, θ) - \frac{{\bar{v}}_{df} \cos θ}{r} \frac{\partial δ {\hat{g}}_{f 0}^{m, n} (r, θ)}{\partial θ} \\ + i \frac{{\bar{v}}_{df} m \cos θ}{r} δ {\hat{g}}_{f 0}^{m, n} (r, θ) - i k_{r} {\bar{v}}_{df} \sin θ δ {\hat{g}}_{f 0}^{m, n} (r, θ)] \\ = - \sum_{m, n} \{i ω \frac{Z_{f} |e|}{T_{f}} J_{0} F_{Mf} δ {\hat{ϕ}}^{m, n} (r) \\ - i ω_{* f}^{m, n} [1 + η_{f} (\frac{E}{T_{f}} - \frac{3}{2})] \\ \frac{Z_{f} |e|}{T_{f}} J_{0} F_{Mf} δ {\hat{ϕ}}^{m, n} (r)\} . \end{matrix}$

(12)

Equation (11) can be solved as,

$δ {\hat{g}}_{f 0}^{m, n} (r, θ) = δ {\hat{g}}_{f 0}^{m, n} (r) .$

(13)

Substituting equation (13) into equation (12) and taking the bounce average [32, 33], we get:

$\begin{matrix} \sum_{m, n} \{(\frac{\partial}{\partial t} + i {〈ω_{df}〉}_{b}^{m, n}) δ {\hat{g}}_{f 0}^{m, n} \\ + i \{ω - ω_{* f}^{m, n} [1 + η_{f} (\frac{E}{T_{f}} - \frac{3}{2})]\} \\ \frac{Z_{f} |e| δ {\hat{ϕ}}^{m, n}}{T_{f}} J_{0} F_{Mf}\} = 0 . \end{matrix}$

(14)

Here, ${〈ω_{df}〉}_{b}^{m, n} = ω_{* f}^{m, n} L_{n_{f}} EG (s, κ) / {RT}_{f}$ is a toroidal precession frequency and ${〈\dots〉}_{b}$ represents a bounce-averaged value. In a high-aspect-ratio fusion device without considering plasma shift and shaping, $G (s, κ)$ can be written as $G (s, κ) = 2 E (κ^{2}) / K (κ^{2}) - 1 + 4 s [E (κ^{2}) / K (κ^{2}) + κ^{2} - 1]$ [32, 34], where $K (κ^{2})$ and $E (κ^{2})$ are the complete elliptic integral. The variable κ is defined as κ² = (1 - αB₀ + ϵαB₀) /2ϵαB₀. Here, α = μ/E (with the magnetic moment μ and kinetic energy E). The magnitude of the magnetic field is $B = B_{0} (1 - ϵ \cos θ)$ (with the inverse aspect ratio ϵ).

According to equation (14), the expression for the non-adiabatic part of the perturbed trapped fast-ion distribution function can be derived as,

$\begin{matrix} δ g_{f} & = & \sum_{m, n} \frac{ω - ω_{* f}^{m, n} [1 + η_{f} (\frac{E}{T_{f}} - \frac{3}{2})]}{ω - {〈ω_{df}〉}_{b}^{m, n}} \\ \times \frac{Z_{f} |e| δ {\hat{ϕ}}^{m, n} e^{i n ξ - i m θ}}{T_{f}} J_{0} F_{Mf} . \end{matrix}$

(15)

Significantly, the precession motion of fast ions can resonate with ITG mode when $ω_{r} = {〈ω_{df}〉}_{b}^{m, n}$ . Here, ω_r is the real frequency of ITG mode. This implies that fast ions can stabilize ITG mode via wave-particle resonance.

The equilibrium distribution function of fast ions is assumed to be a Maxwellian distribution function [11, 34] $F_{Mf} = c_{1} n_{f 0} {(M_{f} / 2 π T_{f})}^{3 / 2}$ $e^{- M_{f} v^{2} / 2 T_{f}}$ , where $c_{1} = π / \int_{1 / B_{0} (1 + ϵ)}^{1 / B_{0} (1 - ϵ)}$ $B_{0} K (κ^{2}) / \sqrt{2 ϵ α B_{0}} d α$ is the normalized coefficient.

Substituting the Maxwellian distribution function into equation (15), the perturbed density of trapped fast ions is written as,

$\begin{matrix} δ n_{f} = - n_{f 0} \frac{Z_{f} |e| δ ϕ}{T_{f}} - \sum_{m, n} \frac{c_{1} n_{f 0}}{2 π^{1 / 2} T_{f}^{1 / 2}} \\ \frac{ω R}{ω_{* f}^{m, n} L_{n_{f}}} \frac{Z_{f} |e| δ {\hat{ϕ}}^{m, n} e^{i n ξ - i m θ}}{T_{f}} \\ \times \int_{0}^{+ \infty} \int_{1 / B_{0} (1 + ϵ)}^{1 / B_{0} (1 - ϵ)} \frac{{1 - \frac{ω_{* f}^{m, n}}{ω} [1 + η_{f} (\frac{E}{T_{f}} - \frac{3}{2})]} \sqrt{E} e^{- \frac{E}{T_{f}}}}{G (s, κ) (E - E_{0}^{m, n})} \\ J_{0}^{2} (\sqrt{2 b_{f} \frac{E}{T_{f}}}) \frac{B}{\sqrt{1 - α B}} d E d α . \end{matrix}$

(16)

Here, $E_{0}^{m, n} = ω {RT}_{f} / (ω_{* f}^{m, n} L_{n_{f}} G (s, κ))$ . Remarkably, when $E = E_{0 r}^{m, n}$ , trapped fast ions can resonate with ITG mode, where $E_{0 r}^{m, n} = ω_{r} {RT}_{f} / (ω_{* f}^{m, n} L_{n_{f}} G (s, κ))$ .

Combining equations (2), (7) and (16) and employing the ballooning transformation [35] $δ ϕ = \sum_{m} e^{- i m θ} \int_{- \infty}^{+ \infty} e^{i η (y - m)} δ \tilde{ϕ} (η) d η / 2 π$ , the quasi-neutrality condition in equation (1) can be expressed as,

$\begin{matrix} \{λ_{1} + (1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}) [{(\frac{σ}{Ω})}^{2} \frac{\partial^{2}}{\partial η^{2}} + η^{2} \\ + (\frac{ι}{Ω}) (\cos η + s η \sin η) + λ_{2}] \\ + \frac{Z_{f} n_{f 0}}{n_{e 0}} (λ_{3} + {〈λ_{4}〉}_{θ} + {〈λ_{5}〉}_{θ}) \\ + \frac{Z_{f} n_{f 0}}{n_{e 0}} {〈λ_{6}〉}_{θ} η^{2}\} δ \tilde{ϕ} = 0, \end{matrix}$

(17)

where,

$\begin{matrix} λ_{1} = \frac{Ω}{b_{i θ} s^{2} (τ_{1} Ω + 1 + η_{i})}, \\ λ_{2} = \frac{1}{b_{i θ} s^{2}} (\frac{- 1}{τ_{1} Ω + 1 + η_{i}} + b_{i θ}), \\ λ_{3} = \frac{τ_{2} Ω}{b_{i θ} s^{2} (τ_{1} Ω + 1 + η_{i})}, \end{matrix}$

$\begin{matrix} {〈λ_{4}〉}_{θ} = - \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{[Ω + \frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (1 - \frac{3}{2} η_{f} + \hat{E} η_{f})] \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} d \hat{E}〉}_{α}, \\ {〈λ_{5}〉}_{θ} = \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{[Ω + \frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (1 - \frac{3}{2} η_{f} + \hat{E} η_{f})] \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} \hat{E} b_{f θ} d \hat{E}〉}_{α}, \\ {〈λ_{6}〉}_{θ} = \frac{2 c_{1} τ_{2}^{2} Ω}{π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \frac{ρ_{T_{f}}^{2}}{ρ_{T_{i}}^{2}} \\ {〈\int_{0}^{+ \infty} \frac{[Ω + \frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (1 - \frac{3}{2} η_{f} + \hat{E} η_{f})] \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} \hat{E} d \hat{E}〉}_{α}, \end{matrix}$

(18)

with y = sk_θx, σ = ϵ_n/sb_iθτ₁q, ι = 2ϵ_n/b_iθs²τ₁, $ϵ_{n} = L_{n_{e}} / R$ , $b_{i θ} = k_{θ}^{2} ρ_{T_{i}}^{2}$ , τ₁ = T_e/T_i and τ₂ = Z_fT_e/T_f. Ω = ω/ω_*e is the normalized frequency and $\hat{E} = E / T_{f}$ is the normalized energy. $\hat{E_{0}} (s, κ) = - τ_{2} R Ω / (L_{n_{e}} G (s, κ))$ with Ω = Ω_r + iγ. Here, Ω_r is the normalized real frequency and γ is the normalized growth rate. The bracket ${〈\dots〉}_{θ}$ represents a θ - average value. The notation for the angle part of integration is expressed as ${〈Q〉}_{α} = \int_{1 / B_{0} (1 + ϵ)}^{1 / B_{0} (1 - ϵ)} {QB}_{0} K (κ^{2}) / \sqrt{2 ϵ α B_{0}} d α$ , where Q is an arbitrary function of α.

Note that, λ₁ expresses the response of the background electrons. λ₂ comes from the background main ions. λ₃, ${〈λ_{4}〉}_{θ}$ , ${〈λ_{5}〉}_{θ}$ , ${〈λ_{6}〉}_{θ}$ stem from fast ions. λ₃ expresses the adiabatic part of fast ions. All of ${〈λ_{4}〉}_{θ}$ , ${〈λ_{5}〉}_{θ}$ and ${〈λ_{6}〉}_{θ}$ arise from the non-adiabatic part of fast ions. Since the Bessel function $J_{0}^{2} (\sqrt{2 b_{f} \hat{E}})$ is expanded as $J_{0}^{2} (\sqrt{2 b_{f} \hat{E}}) \approx (1 - \hat{E} b_{f θ}) + \hat{E} ρ_{T_{f}}^{2} \partial^{2} / \partial x^{2}$ , ${〈λ_{4}〉}_{θ}$ stems from the principal part of the Bessel function. ${〈λ_{5}〉}_{θ}$ , ${〈λ_{6}〉}_{θ}$ come from the poloidal component and the radial component of the Bessel function, respectively. Namely, ${〈λ_{5}〉}_{θ}$ and ${〈λ_{6}〉}_{θ}$ denote the finite Larmor radius effect.

To facilitate analysis, ${〈λ_{4}〉}_{θ}$ is divided into ${〈λ_{41}〉}_{θ}$ , ${〈λ_{42}〉}_{θ}$ , ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ , i.e. ${〈λ_{4}〉}_{θ} = {〈λ_{41}〉}_{θ} + {〈λ_{42}〉}_{θ} + {〈λ_{43}〉}_{θ} + {〈λ_{44}〉}_{θ}$ ,

$\begin{matrix} {〈λ_{41}〉}_{θ} = - \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{Ω \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} d \hat{E}〉}_{α}, \\ {〈λ_{42}〉}_{θ} = - \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{\frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} d \hat{E}〉}_{α}, \\ {〈λ_{43}〉}_{θ} = - \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{\frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (- \frac{3}{2} η_{f}) \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} d \hat{E}〉}_{α}, \\ {〈λ_{44}〉}_{θ} = - \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{3}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \\ {〈\int_{0}^{+ \infty} \frac{\frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (\hat{E} η_{f}) \sqrt{\hat{E}} e^{- \hat{E}}}{G (s, κ) (\hat{E} - \hat{E_{0}} (s, κ))} d \hat{E}〉}_{α} . \end{matrix}$

(19)

Corresponding to equation (4), ${〈λ_{41}〉}_{θ}$ implies the effect of the energy-gradient-driven term of fast ions. ${〈λ_{42}〉}_{θ}$ represents the effect of the density-gradient-driven term. Both of ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ stem from the temperature-gradient-driven terms and are denoted simply as the - 3η_f/2 term and $\hat{E} η_{f}$ term. They are opposite in sign, which signifies different effects between them. The ordering relationship between ${〈λ_{42}〉}_{θ}$ , ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ is about $1 : 3 η_{f} / 2 : {\hat{E}}_{0 r} η_{f}$ . The same division is applied equally to ${〈λ_{5}〉}_{θ}$ and ${〈λ_{6}〉}_{θ}$ .

Following reference [4] and proceeding to perform the strong coupling approximation $\cos η + s η \sin η = 1 + (s - 1 / 2) η^{2}$ , equation (17) can be written as,

$\begin{matrix} \{λ_{1} + (1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}) [{(\frac{σ}{Ω})}^{2} \frac{\partial^{2}}{\partial η^{2}} \\ + η^{2} + (\frac{ι}{Ω}) (1 + (s - \frac{1}{2}) η^{2}) + λ_{2}] \\ + \frac{Z_{f} n_{f 0}}{n_{e 0}} (λ_{3} + {〈λ_{4}〉}_{θ} + {〈λ_{5}〉}_{θ}) \\ + \frac{Z_{f} n_{f 0}}{n_{e 0}} {〈λ_{6}〉}_{θ} η^{2}\} δ \tilde{ϕ} = 0, \end{matrix}$

(20)

We find that equation (20) is just the familiar Weber–Hermite equation, as shown in [4, 36]. The eigenfunction solution is the Hermite function. Considering only the lowest eigenstate and seeking the solution of the form $δ \tilde{ϕ} = \exp (- ζ η^{2})$ , the dispersion equation is obtained:

$\begin{matrix} b_{s} {(\frac{q Ω}{ϵ_{n}})}^{2} {(\frac{2 ϵ_{n}}{Ω} + \frac{b_{s} s^{2} \bar{λ}}{1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}})}^{2} + b_{s} s^{2} \\ + (2 s - 1) \frac{ϵ_{n}}{Ω} + \frac{\frac{Z_{f} n_{f 0}}{n_{e 0}} {〈λ_{6}〉}_{θ} b_{s} s^{2}}{1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}} = 0, \end{matrix}$

(21)

with,

$\begin{matrix} 2 ζ = b_{s} {(\frac{q Ω}{ϵ_{n}})}^{2} [\frac{2 ϵ_{n}}{Ω} + \frac{b_{s} s^{2} \bar{λ}}{1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}}], \\ \bar{λ} = λ_{1} + (1 - \frac{Z_{f} n_{f 0}}{n_{e 0}}) λ_{2} + \frac{Z_{f} n_{f 0}}{n_{e 0}} (λ_{3} + {〈λ_{4}〉}_{θ} + {〈λ_{5}〉}_{θ}) . \end{matrix}$

(22)

Here, b_s = τ₁b_iθ. Distinctly, let n_f0 = 0 and equation (21) returns to the eigenvalue equation of ITG mode in [4].

It is important to note that a significant fraction of trapped fast ions can resonate with ITG mode when the precession frequency of fast ions is close to the frequency of ITG mode, i.e. when $\hat{E} = {\hat{E}}_{0 r} (s, κ)$ . Here, ${\hat{E}}_{0 r} (s, κ) = - τ_{2} R Ω_{r} / (L_{n_{e}} G (s, κ))$ suggests the normalized energy of fast ions that resonate with ITG mode at a frequency Ω_r. Distinctly, the resonant condition depends on τ₂, i.e. T_f/T_e. According to equation (18), the contributions to the non-adiabatic part of fast ions stem from the energy, density and temperature gradients of fast ions. If η_f ≫ 1, the temperature-gradient-driven terms ( ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ ) will be dominant since the ordering relationship between ${〈λ_{42}〉}_{θ}$ , ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ is about $1 : 3 η_{f} / 2 : {\hat{E}}_{0 r} η_{f}$ . Subsequently, there is a threshold condition between ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ . At relatively low temperature, the energy of resonant fast ions, i.e. ${\hat{E}}_{0 r}$ is relatively high. Therefore, ${\hat{E}}_{0 r} η_{f} > 3 η_{f} / 2$ . When the temperature of fast ions increases, the resonant energy ${\hat{E}}_{0 r}$ decreases and then ${\hat{E}}_{0 r} η_{f} < 3 η_{f} / 2$ . The threshold condition implies that wave-fast-ion resonance may play different roles (stabilizing or destabilizing) in ITG mode at different temperatures.

3. Numerical results

In order to obtain further understanding of fast-ion stabilization on ITG instability, the main physical parameters including the density n_f0/n_e0, temperature T_f/T_e, density gradient $L_{n_{e}} / L_{n_{f}}$ and the ratio of the temperature and density gradients η_f of fast ions are investigated in more detail. Significantly, the effects of the background-driven terms (energy and space gradients) are studied separately.

In numerical calculation, equation (21) is solved without any approximation. The plasma parameters are mainly taken from a JET L-mode discharge 73 224 [22, 28] and are listed in table 1. To facilitate the analysis, there is a single fast particle species, fast helium-3, presented in the calculation and the bulk plasma is composed of deuterium and electrons.

Table 1. Parameters for the JET discharge 73 224 with fast helium.

Parameter	$r (m)$	$a (m)$	R/a	q	s	$B_{0} (T)$	$T_{e} (keV)$	T_i/T_e
Value	0.36	0.96	3.12	1.74	0.523	3	3.2	1.0
Parameter	n_f0/n_e0	R/L_{n_i}	R/L_{T_i}	η_i	R/L_{n_f}	R/L_{T_f}	η_f	m
Value	0.07	1.3	9.3	7.15	1.6	23.1	14.4	26

| Show Table

DownLoad: CSV

3.1 Effects of fast-ion density, n_f0/n_e0

In this subsection, the effects of n_f0/n_e0 on both growth rate γ and real frequency Ω_r of ITG mode are presented in figures 1(a) and (b), respectively. The temperatures of fast helium are at T_f = 5T_e and T_f = 30T_e.

Figure 1. (a) Normalized growth rate and (b) normalized real frequency of ITG mode with different densities of fast ions. Red and black dashed lines show the case without fast ions and the dilution, respectively. Solid blue and orange lines show the case with fast helium at T_f = 5T_e and T_f = 30T_e.

DownLoad: Full-Size Img PowerPoint

From figure 1(a), it is found that fast ions play a stabilizing role on ITG mode and the growth rate reduces with the increasing n_f0/n_e0. As shown by the solid blue line in figure 1(a), relative to dilution, fast ions destabilize at T_f = 5T_e. The dominant effect of fast ions is dilution at T_f = 5T_e. However, the growth rate is lower than dilution at T_f = 30T_e. In addition, as shown in figure 1(b), the dilution leads to a reduction in the real frequency of ITG mode. Oppositely, the real frequencies of the two cases with fast helium rise as n_f0/n_e0 increases. A negative value of the real frequency Ω_r signifies a mode propagating in the ion direction.

The main physical mechanisms can be simply explained. When positively-charged fast ions are added to the background plasmas, the electromagnetic fields will have less response to the main thermal ions [14, 18]. Consequently, the growth rate arising from the bulk ions is reduced. Compared to the pure dilution case, both the growth rates and real frequencies in the two cases with fast helium are different. This fact demonstrates that there is another kinetic effect of fast ions in addition to dilution. Furthermore, the two cases at different fast-ion temperatures imply that the kinetic effect depends on the temperature of fast ions. Distinctly, this is the resonance mechanism that conforms to the last analysis in section 2. At low temperature (T_f = 5T_e), two temperature-gradient-driven terms are assumed ${\hat{E}}_{0 r} η_{f} > 3 η_{f} / 2$ and wave-particle resonance plays a destabilizing role. However, at relatively high temperature (T_f = 30T_e), ${\hat{E}}_{0 r} η_{f} < 3 η_{f} / 2$ and resonance leads to stabilization of ITG mode. Moreover, the resonant effect is weak at low temperature since only a small fraction of fast ions can resonate with ITG mode. As the temperature of fast ions rises, the fraction of resonant fast ions increases. Thus, the resonant effect of fast ions in the case at T_f = 30T_e is stronger than that at T_f = 5T_e.

3.2 Effects of fast-ion temperature, T_f/T_e

In this subsection, the effects of T_f/T_e on both growth rate γ and real frequency Ω_r of ITG mode are given in figures 2(a) and (b), respectively. Subsequently, the results are explained in detail using figures 3–5. The density of fast ion is n_f0/n_e0 = 0.07.

Figure 2. ITG (a) normalized growth rate and (b) normalized real frequency as a function of T_f/T_e.

DownLoad: Full-Size Img PowerPoint

Figure 3. ITG normalized growth rate as a function of T_f/T_e: (a) adiabatic response, (b) response of energy-gradient-driven term, (c) sum of adiabatic response and response of energy-gradient-driven term and (d) response of space-gradient-driven term.

DownLoad: Full-Size Img PowerPoint

Figure 4. Imaginary part of fast ions driven by (a) density-gradient term (

${〈λ_{42}〉}_{θ}$ ), (b) temperature-gradient

$- \frac{3}{2} η_{f}$ term (

${〈λ_{43}〉}_{θ}$ ) and (c) temperature-gradient

$\hat{E} η_{f}$ term (

${〈λ_{44}〉}_{θ}$ ).

DownLoad: Full-Size Img PowerPoint

Figure 5. Imaginary part of fast ions: (a) principal part of Bessel function (

${〈λ_{4}〉}_{θ}$ ), (b) finite Larmor radius effect in θ direction (

${〈λ_{5}〉}_{θ}$ ) and (c) finite Larmor radius effect in r direction (

${〈λ_{6}〉}_{θ}$ ).

DownLoad: Full-Size Img PowerPoint

In figure 2(a), it is found that, for T_f/T_e < 7, wave-fast-ion resonance leads to destabilization of ITG mode and the dominant effect is dilution. When the temperature of fast ions exceeds a critical value (T_f/T_e ~ 7), the growth rate decreases with the increasing T_f/T_e. Meanwhile, resonance plays a stabilizing role on ITG mode. At higher temperature, i.e. T_f/T_e > 30, as the temperature rises, the growth rate changes slowly. In figure 2(b), it can be observed that, as T_f/T_e rises, the real frequency Ω_r increases and then moves towards the frequency in the dilution case.

First, to understand the results, in figures 3(a)–(d) both the adiabatic response and non-adiabatic response driven by energy and space gradients are depicted separately. In figure 3(a), the adiabatic response is shown by solving equation (21) with λ₃ of fast ions only, namely, let ${〈λ_{4}〉}_{θ}$ , ${〈λ_{5}〉}_{θ}$ and ${〈λ_{6}〉}_{θ}$ be zero. As can be seen, the adiabatic part of fast ions stabilizes the ITG, but this stabilizing effect quickly decays to zero as the temperature increases. The effect of the energy-gradient-driven term is shown in figure 3(b) by retaining ${〈λ_{41}〉}_{θ}$ , ${〈λ_{51}〉}_{θ}$ and ${〈λ_{61}〉}_{θ}$ for fast ions only. In contrast to the adiabatic response, the energy-gradient-driven term destabilizes the ITG mode. Similarly the destabilizing effect also rapidly weakens to zero with the increasing T_f/T_e. Significantly, as shown in figure 3(c), the effects of the adiabatic part and energy-gradient-driven term of fast ions almost cancel each other so that the effect of fast ions mainly results from the space-gradient-driven term, which is shown in figure 3(d).

Second, the effects of the fast-ion space gradient including density and temperature gradients are investigated in more detail. In figure 4(a), the magnitude of the imaginary part of ${〈λ_{42}〉}_{θ}$ , which represents the effect of density gradients, is shown. Here, to facilitate analysis, a certain angle is assumed which satisfies κ² = 0.6. The imaginary part of $- {〈λ_{42}〉}_{θ}$ is basically positive, which implies that the density-gradient term plays a destabilizing role on ITG mode. Similarly, $- Im ({〈λ_{44}〉}_{θ})$ in figure 4(c) is also mainly positive, which suggests that one part of temperature-gradient-driven terms ( $\hat{E} η_{f}$ term) is destabilizing on ITG mode. However, as shown in figure 4(b), another part (- 3η_f/2 term) plays a stable role on ITG since $- Im ({〈λ_{43}〉}_{θ})$ is mostly negative. Distinctly, both $|Im ({〈λ_{43}〉}_{θ})|$ and $|Im ({〈λ_{44}〉}_{θ})|$ are much larger than $|Im ({〈λ_{42}〉}_{θ})|$ , which conforms to the ordering relationship between ${〈λ_{42}〉}_{θ}$ , ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ , i.e. $1 : 3 η_{f} / 2 : {\hat{E}}_{0 r} η_{f}$ . Therefore, the temperature gradient of fast ions expressed by ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ is mainly responsible for the kinetic effect of fast ions on ITG mode.

Third, the resonant fast-ion stabilizing mechanism is studied. As shown in figures 4(b) and (c), at relatively low temperature, the magnitude of both $|Im ({〈λ_{44}〉}_{θ})|$ and $|Im ({〈λ_{43}〉}_{θ})|$ is small. As T_f/T_e rises, the magnitude of $|Im ({〈λ_{44}〉}_{θ})|$ and $|Im ({〈λ_{43}〉}_{θ})|$ increases and becomes maximum around T_f ~ 12T_e. Then, the magnitude of $|Im ({〈λ_{44}〉}_{θ})|$ and $|Im ({〈λ_{43}〉}_{θ})|$ decreases with increasing T_f/T_e. This result conforms to the analysis of resonance. When the temperature of fast ions is low, only a small fraction of fast ions can resonate with ITG mode. Therefore, the resonant effect of fast ions is weak at relatively low temperature. As T_f/T_e rises, the fraction of resonant fast ions increases, which implies that the fast-ion resonant effect strengthens. When the temperature exceeds a certain value (T_f ~ 12T_e), the fraction of resonant fast ions decreases and the resonant effect weakens with increasing T_f/T_e. In addition, at relatively low temperature, $|Im ({〈λ_{44}〉}_{θ})| > |Im ({〈λ_{43}〉}_{θ})|$ , which suggests the destabilizing effect of the $\hat{E} η_{f}$ term is stronger than the stabilizing effect of the - 3η_f/2 term. When the temperature exceeds a critical value, conversely, the stabilizing effect of the -3η_f/2 term is stronger. This result conforms to the last analysis of resonance in section 2. As the fast-ion temperature increases, ${\hat{E}}_{0 r}$ decreases. When the threshold condition ${\hat{E}}_{0 r} < 3 / 2$ is satisfied, stabilization of ITG mode by fast ions realized.

Finally, the finite Larmor radius effect is studied in figures 5(a)–(c). In figure 5(a), as the principal part from the expansion of $J_{0}^{2}$ , for low temperatures, $- Im ({〈λ_{4}〉}_{θ})$ is positive, which suggests the destabilizing effect. When the temperature exceeds a certain value, $- Im ({〈λ_{4}〉}_{θ})$ becomes negative and fast ions stabilize ITG via wave-fast-ion resonance. Significantly, as shown in figures 5(b) and (c), the finite Larmor radius effect stabilizes ITG mode since both $- Im ({〈λ_{5}〉}_{θ})$ and $- Im ({〈λ_{6}〉}_{θ})$ are negative.

3.3 Effects of fast-ion density gradient $L_{n_{e}} / L_{n_{f}}$ and η_f

In this subsection, the effects of η_f and $L_{n_{e}} / L_{n_{f}}$ on both growth rate γ and real frequency Ω_r of ITG mode are presented in figures 6(a)–(d) separately. The density of fast ions is n_f0/n_e0 = 0.07 and the temperature of fast ions is T_f = 30T_e. In figures 6(a) and (b), the density gradient of fast ions remains unchanged, i.e. $R / L_{n_{f}} = 1.6$ . In figures 6(c) and (d), η_f is fixed, i.e. η_f = 14.4.

Figure 6. ITG (a) normalized growth rate and (b) real frequency as a function of η_f. ITG (c) normalized growth rate and (d) real frequency versus

$L_{n_{e}} / L_{n_{f}}$ .

DownLoad: Full-Size Img PowerPoint

First, from figures 6(a) and (c), it is observed that both increasing η_f and $L_{n_{e}} / L_{n_{f}}$ will reduce the growth rate of ITG mode, which implies that the increasing η_f and $L_{n_{e}} / L_{n_{f}}$ will strengthen the stabilizing effect of fast ions. Second, from figure 6(b), it is found that, as η_f increases, the real frequency of ITG mode increases and then moves towards the frequency in the dilution case. Finally, as shown in figure 6(d), the real frequency of ITG mode rises with increasing $L_{n_{e}} / L_{n_{f}}$ .

The results in figures 6(a) and (c) can be simply explained. At first, in section 3.2, it is shown that fast-ion resonance leads to the stabilization of ITG mode at T_f = 30T_e. Both $L_{n_{e}} / L_{n_{f}}$ and η_f are unrelated to the resonance condition. Therefore, wave-particle resonance always plays a stable role at different η_f and $L_{n_{e}} / L_{n_{f}}$ . In equation (18), it can be seen that the temperature-gradient-driven terms of fast ions are proportional to η_f, i.e. ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ are proportional to η_f. Thus, the increasing η_f will increase fast-ion resonant stabilization. From equation (18), it is also found that the space-gradient-driven terms including density- and temperature-gradient-driven terms are also proportional to $L_{n_{e}} / L_{n_{f}}$ , namely, ${〈λ_{42}〉}_{θ}$ , ${〈λ_{43}〉}_{θ}$ and ${〈λ_{44}〉}_{θ}$ are proportional to $L_{n_{e}} / L_{n_{f}}$ . Therefore, the stabilizing effect of fast ions strengthens as $L_{n_{e}} / L_{n_{f}}$ increases.

In summary, both the increasing η_f and $L_{n_{e}} / L_{n_{f}}$ can strengthen the stabilization of fast ions on ITG mode. Therefore, there are two ways to improve the stabilizing effect of fast ions on ITG mode. First, keep $L_{n_{e}} / L_{n_{f}}$ and increase the ratio of the temperature and density gradients of fast ions, i.e. increase η_f. Second, keep η_f and increase the density gradient of fast ions, i.e. increase $L_{n_{e}} / L_{n_{f}}$ .

3.4 Influence of resonance broadening of wave-particle interaction

The influence of resonance broadening of wave-particle interaction is discussed in this subsection. As described above, the resonance condition is written as $\hat{E} = {\hat{E}}_{0 r}$ , i.e. $ω_{r} = {〈ω_{df}〉}_{b}^{m, n}$ . However, the resonance is broadened due to the growth rate of ITG mode.

The growth rate of ITG mode without fast ions is denoted as γ_ITG here. Assuming γ_ITG = 0, and the wave-particle resonance contribution without resonance broadening is then evaluated. The response of the resonant fast ions is calculated by replacing ${〈λ_{4}〉}_{θ}$ with res ${〈λ_{4}〉}_{θ}$ . Following Landau's prescription and res ${〈λ_{4}〉}_{θ}$ is given by,

$\begin{matrix} res {〈λ_{4}〉}_{θ} = i \frac{2 c_{1} τ_{2}^{2} Ω}{b_{i θ} s^{2} π^{\frac{1}{2}} ϵ_{n} (τ_{1} Ω + 1 + η_{i})} \frac{1}{G (s, κ)} \\ \sqrt{\hat{E_{0 r}}} e^{- \hat{E_{0 r}}} [Ω_{r} + \frac{L_{n_{e}}}{τ_{2} L_{n_{f}}} (1 - \frac{3}{2} η_{f} + \hat{E_{0 r}} η_{f})] \frac{B_{0} K (κ^{2})}{\sqrt{2 ϵ α B_{0}}} . \end{matrix}$

Here, the same angle κ² = 0.6 is also assumed.

In figure 7, the magnitude of res ${〈λ_{4}〉}_{θ}$ is depicted as a function of T_f/T_e. It is found that wave-particle resonance plays a stabilizing role on ITG mode since $- Im (res {〈λ_{4}〉}_{θ})$ is negative. As T_f/T_e increases, the magnitude of $- Im (res {〈λ_{4}〉}_{θ})$ decreases rapidly and the fast-ion resonance stabilization effect weakens. At high temperature, i.e., T_f ~ 40T_e, the resonance effect is probably negligible, which implies that fast ions may be modeled by pure dilution [17].

Figure 7. Imaginary part of fast ions from wave-fast-ion resonance as a function of T_f/T_e.

DownLoad: Full-Size Img PowerPoint

Comparing figure 5(a) with figure 7, it is found that the resonance broadening (from γ_ITG) actually reduces the resonance effect since the magnitude of $- Im ({〈λ_{4}〉}_{θ})$ is smaller than $- Im (res {〈λ_{4}〉}_{θ})$ . Moreover, in figure 5(a), at T_f/T_e > 30, $- Im ({〈λ_{4}〉}_{θ})$ regains slowly with increasing T_f/T_e and the magnitude will not reduce to zero at high temperature (T_f ~ 40T_e). The strong resonance broadening effect is the main reason for the difference between our results and the simulation ones [22]. The expression of ${〈λ_{4}〉}_{θ}$ in equation (18) can be used to explain this result. Fast ions can resonate with ITG mode when $\hat{E} = {\hat{E}}_{0 r} (s, κ)$ . However, the denominator of ${〈λ_{4}〉}_{θ}$ is not zero since there is an imaginary part, i.e. $i {\hat{E}}_{0 i}$ . This imaginary part makes a difference to the resonance effect. In addition, the growth rate (γ_ITG) makes a difference to the adiabatic response of fast ions. When γ_ITG = 0, the adiabatic part of fast ions no longer contributes an imaginary part and cannot offset the effect of the energy-gradient-driven term.

In summary, the growth rate of ITG mode brings resonance broadening which reduces the effect of the resonance and makes a difference. As a result, the stabilizing effect of fast ions on ITG mode is underestimated in our work, since γ_ITG in our background model is much larger than the actual growth rate of ITG mode.

4. Conclusion

The stabilizing effect of trapped fast ions on ITG mode has been studied, based on both analytical and numerical calculations. The relevant physics mechanisms have been explained.

It is found that fast ions can strongly affect ITG mode through a wave-particle resonance mechanism when the precession frequency of trapped fast ions is close enough to the frequency of ITG mode. The fast-ion stabilizing effect depends on density, temperature, and the density and temperature gradients of fast ions.

Fast-ion resonance destabilizes ITG mode at very low temperature, but is stabilizing as soon as the fast-ion temperature exceeds a certain value. By investigating the effect of the fast-ion temperature in more detail, it is found that the effects of the adiabatic part and energy-gradient-driven term of fast ions almost cancel each other. Thus, the effect of fast ions mainly results from the space-gradient-driven term. The space-gradient-driven term is derived from density and temperature gradients. The density-gradient term of fast ions plays a destabilizing role on ITG mode. Moreover, one part of the temperature-gradient-driven term ( $\hat{E} η_{f}$ term) is destabilizing on ITG mode, but another part (-3η_f/2 term) plays a stable role on ITG. When the threshold condition ${\hat{E}}_{0 r} < 3 / 2$ is satisfied, stabilization of ITG mode by fast ions is realized. Increasing the density of fast ions can enhance their effects. In addition, both increasing η_f and $L_{n_{e}} / L_{n_{f}}$ can strengthen the fast-ion resonant stabilization effect.

These findings contribute to the understanding of stabilization of ITG mode by trapped fast ions and suggest a means for improving ion energy confinement in fusion devices. However, in our analytic model, the growth rate of ITG mode without fast ions, i.e. γ_ITG is large since the resonant effect of background main ions is ignored. This large γ_ITG weakens the wave-fast-ion resonant effect in our work. The improvement of this issue will be shown in our future work. In addition, the electromagnetic effect is different to the resonance effect. It will be complex if the mode coupling effect is considered. In this work, the electromagnetic effect is not considered, which is left for future research.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos. 11822505, 11835016 and 11675257), the Youth Innovation Promotion Association CAS, the Users with Excellence Program of Hefei Science Center CAS (No. 2019HSC-UE013), the Fundamental Research Funds for the Central Universities (No. WK3420000008) and the Collaborative Innovation Program of Hefei Science Center CAS (No. 2019HSC-CIP014).

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Article Contents

Abstract

1. Introduction

2. Dispersion relation

3. Numerical results

4. Conclusion

Acknowledgments

References

Stabilization of ion-temperature-gradient mode by trapped fast ions