Enhancement of neoclassical impurity density up/down asymmetry and Pfirsch–Schlüter transport due to the plasma elongation in the tokamak plasmas

The standard neoclassical theory of the impurity transport in tokamak plasma is improved by including the plasma elongation effect. The neoclassical up/down asymmetry of the impurity density is largely enhanced by the plasma elongation and is under-predicted by the neoclassical theory with the circular cross section assumption. Accordingly, the impurity Pfirsch–Schlüter neoclassical transport is also enhanced by the plasma elongation.


Introduction
Understanding the impurity transport in tokamak plasmas is important for the currently operated tokamaks and future fusion devices, such as ITER. Impurity accumulation in the core of tokamak plasmas is well known to be a big challenge due to fuel dilution and power loss from radiation. This has been an issue of great concern in the magnetic confinement fusion research field. Reduction or suppression of the core impurity accumulation is essential to realize a magnetic confinement fusion reactor.
Both the neoclassical and turbulent transports can play important roles in determining the impurity concentration and the impurity density profile [1,2]. The relative role depends on the plasma parameters and the impurity species. For light Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. impurities, turbulent transport can dominate over the neoclassical transport in tokamak core plasmas. While for the heavy impurities, the neoclassical transport could compete with or even dominate over the turbulent transport. Additionally, the magnetohydrodynamic (MHD) activity could also impact the impurity transport [2]. Admittedly, the neoclassical transport plays an important role in the impurity particle transport [1,2]. The core impurity accumulation is mainly driven by the neoclassical inward particle convective transport, which is supported by the experimental observations [3][4][5][6][7].
In the tokamak plasmas the impurity ions are actually not distributed uniformly on the magnetic flux surface [8][9][10], especially for the heavy impurity ions. While the poloidally asymmetric impurity density distribution on the magnetic flux surface can significantly affect the impurity neoclassical radial particle transport [11][12][13][14]. The impurity neoclassical radial particle flux is proportional to the impurity density up/down asymmetry [15,16]. The magnitude and direction (inward or outward) of the impurity neoclassical particle flux are strongly related to the magnitude and direction (up-down or down-up) of the impurity density up/down asymmetry.
The up/down asymmetry of the impurity density was observed in the different tokamak experiments, such as in the Alcator [17], PLT [18], PDX [19], ASDEX [20], COMPASS-C [21], and C-Mod [22][23][24][25] tokamak plasmas. The neoclassical theory [16,26] was proposed to explain this kind of experimental phenomenon. The neoclassical theory can predict the direction of the asymmetry observed in the experiments, but the predicted asymmetry magnitude is much smaller than the experimental measurements [23,25]. The prediction of the conventional neoclassical theory is based on the circular cross section assumption for the tokamak plasma, while the current tokamaks and the future fusion reactors are usually operated with the elongated plasmas. In this paper, the effect of the plasma elongation on the neoclassical impurity density up/down asymmetry and the radial particle transport will be investigated.
The remaining part of this paper is organized as follows. In section 2, the theoretical model will be presented. In section 3, the effect of the plasma elongation on the impurity density neoclassical up/down asymmetry is investigated. The effect of the plasma elongation on the impurity neoclassical radial particle transport will be discussed in section 4. In section 5, the summary and conclusion will be presented.

Theoretical model
The neoclassical up/down asymmetry of the impurity density in a non-rotating tokamak plasma is determined by the impurity parallel momentum equation (equation (7) in [27]), where B is the magnetic field, p Z is the impurity thermal pressure, F Zi is the friction force between the impurity ions and the main ions, which is written in the following form, where m i and n i and are the mass and density of the main ions, is the ion-impurity collision time, I = R 0 B φ 0 is a function of the magnetic flux surface, R 0 and B φ 0 are the major radius and the toroidal magnetic field at the magnetic axis respectively, ψ is the poloidal magnetic flux and T i is the main ion temperature, B φ is the toroidal magnetic field, K Z and u are both the flux function and related with the poloidal rotation of the impurity and main ions, respectively. (r, θ, φ ) will be chosen as the right-hand coordinate system in this paper. r labels the minor radius at the low-field-side middle plane of the concerned magnetic flux ψ ; θ labels the poloidal angle in the flux surface cross section and increases in the counter-clockwise direction, and θ = 0 corresponds to the low-field-side middle plane; φ is the toroidal angel of the torus. L −1 ⊥ is defined as, where k is the coefficient of the ion thermal force due to the ion temperature gradient and depends on the collisionality regimes [28]. Here, we will not discuss how to determine k and just take it as a known value.
To obtain equation (2), the lower-order main ion and impurity flow within the magnetic flux surface in the following are used, The main ion thermal force is included in the friction force and u i// (equation (5) in [29]) in equation (2) is where ⟨⟩ denotes the flux-surface-averaged. For the trace impurities, the impurity parallel viscosity force is n Z Z 2 /n i (≪1) smaller compared with the ion-impurity friction force [30]. Therefore, the impurity parallel viscosity is neglected in equation (1). The electron-impurity and impurityimpurity friction forces are √ m e /m i and √ m Z n Z /m i n i smaller compared with the ion-impurity friction force, respectively, and are also neglected [26]. Also the effect of the poloidal electric field [26], which is dominated by the main ion dynamics due to the plasma rotation [9] or the ion cyclotron resonance heating (ICRH) and electron cyclotron resonance heating (ECRH) [31], is not included in equation (1). We will focus on the main effects of the plasma elongation.
With the lowest order in/out and up/down asymmetries retained, the impurity density poloidal variation can be decomposed as n Z = n Z0 (1 + n Zs sin θ + n Zc cos θ) .
Combining equations (1) and (6), the fundamental equation to derive the neoclassical impurity density up/down asymmetry can be obtained In equation (7), the magnetic field B = B 0 R 0 /h φ and the poloidal magnetic field B θ = B θ0 R 0 / (h φ h r ) have been used [32]. h r ,h θ and h φ are the metric coefficients for (r, θ, φ ) coordinate system. Here, B θ0 is the θ-independent poloidal magnetic field.
By integration equation (7) over 2π , we can obtain where By multiplying equation (7) with sin θ and cos θ and then integrating over 2π, we can obtain where G = B0 With the help of equation (8), the neoclassical up/down asymmetry of the impurity density can be derived from equations (9) and (10), Equation (11) is the impurity density neoclassical up/down asymmetry in a tokamak plasma including the effects of the plasma shape and Shafranov shift, which are included in the To calculate the neoclassical impurity density up/down asymmetry in an elongated tokamak plasma, the Miller equilibrium model is employed to derive the metric coefficient integrations in equation (11). Following the Miller equilibrium model [33], the shape of the magnetic flux surface for an elongated tokamak plasma without the triangularity and Shafranov shift in the (R, Z) coordinates can be specified by where r is the minor radius of the magnetic flux surface at the equatorial plane, θ is the poloidal angle, and κ is the plasma elongation. Then the metric coefficients will be (equation (9) in [32]) Here, ε = r/R 0 . Based on these metric coefficient integrations in equation (11) can be obtained. Then, the effects of the plasma elongation on the neoclassical impurity density up/down asymmetry and radial particle transport can be investigated.

From the definitions of the metric coefficient integrations (equations (14)-(16)), it is easily found that
J0 H 1 ≪ H 3 and J1 J0 J 1 ≪ J 3 in the elongated tokamak plasma. Then equation (8) and equation (11) will be reduced to The metric coefficient integrations in equation (18) can be obtained analytically, Then, the neoclassical up/down asymmetry of the impurity density in an elongated tokamak plasma can be ultimately derived, The second term in the denominator in equation (20) is due to the coupling of the in/out and up/down asymmetries of the impurity density through the ion-impurity friction force. While at the current typical tokamak edge, where the plasma has larger elongation, such as n i ∼ 10 19 m −3 , T i ∼ 1.0keV, the safety factor q ∼ 3.0, the impurity poloidal rotation speed v Zθ = KZB θ0 nZ0 ∼ 1 km s −1 , Gκr KZB0 nZ0 will be much less than 1 for the light and medium impurities and for the non-fully stripped heavy impurities. Then, equation (20) can be simplified further, This is consistent with the conventional neoclassical theory under the circular cross section assumption, i.e. with κ = 1 as shown in section IV.A in [16]. The radial gradient of the poloidal magnetic flux in equation (21) is affected by the plasma elongation and needs to be determined. Here it should be pointed out that the factor G in equation (21) contains B θ0 , which is affected by the plasma elongation and will be discussed in the following.
From the definition of the poloidal magnetic field, we can obtain While the poloidal magnetic field can also be determined through the Ampere's law, where dl θ = h θ dθ, I P (r) is the total plasma current inside the minor radius r. Then B θ0 can be derived whereB θ0 = µ0IP(r) 2π r is the θ-independent poloidal magnetic field with κ = 1 . The θ-independent poloidal magnetic field is decreased for the elongated tokamak plasmas with the plasma current fixed. This is the main reason why the plasma elongation affects the neoclassical impurity density up/down asymmetry, which will be discussed in the following. The factor G will affect the impurity density up/down asymmetry through B θ0 .
By combining equations (22) and (24), the radial gradient of the poloidal magnetic flux can be derived, Then the neoclassical impurity density up/down asymmetry in the elongated tokamak plasmas will be n Zs = n κ=1 where n κ=1 is the neoclassical impurity density up/down asymmetry with κ = 1. The effect of the plasma elongation on the neoclassical impurity density up/down asymmetry is shown in figure 1. With the plasma elongation κ = 2, the plasma elongation magnifies the neoclassical impurity density up/down asymmetry by the factor about three. This enhancement is mainly due to the weakness of the θ-independent poloidal magnetic field by the plasma elongation with the plasma current fixed as mentioned above. The impurity density up/down asymmetry is largely underpredicted by the neoclassical theory with the circular cross section assumption.
In [23], the impurity density up/down asymmetry from the standard neoclassical theory prediction is less than the experimental measurement. For the Alcator C-Mod plasma, κ = 1.8, the improved theory with the plasma elongation effect included predicts the up/down asymmetry of the impurity density as n Zs = 2.5n κ=1 , which is 2.5 times larger than the standard neoclassical theory prediction. A large experimentally measured up/down asymmetry of the impurity density (ñ Z ⟨nZ⟩ ∼ 0.6 in [23]) in the Alcator C-Mod edge plasma will be predicted with m i n i ∼ 0.4, which is reasonable for the Alcator C-Mod edge plasma [24]. While the standard neoclassical theory only predicted the maximum impurity density up/down asymmetry to be 0.3 even with the very large (g ≫ 1 in [23]).

Elongation on the impurity neoclassical particle transport
The impurity Pfirsch-Schlüter neoclassical particle transport can be derived from the toroidal component of the impurity momentum equation [1], The impurity neoclassical radial particle transport is proportional to the impurity density up/down asymmetry, which is determined by equation (26). Then we can obtain The impurity neoclassical radial particle flux increases due to the plasma elongation by the factor ( 1+κ 2 2κ ) 2 , as is shown in figure 2. With the plasma elongation κ = 3, the impurity neoclassical radial particle flux will be enhanced by a factor of about 2.8. The enhancement of the impurity density up/down asymmetry by the plasma elongation enhances the impurity neoclassical radial particle transport, while the plasma elongation itself reduces the impurity neoclassical radial particle transport due to the increasing of the magnetic flux surface area. The impurity neoclassical particle transport in the elongated tokamak plasmas will be underestimated by the neoclassical theory with the circular plasma assumption.
The improved theory is compared with the first-principle neoclassical transport code NEO [34][35][36]. The impurity neoclassical radial particle flux calculated with NEO versus the plasma elongation is also shown in figure 2. In the NEO calculations the plasma profiles are fixed. The plasma elongation is varied and at the same time ψ ′ is varied as equation (25) in the equilibrium. The results from the theory, i.e. equation (28), agree well with the NEO calculations.

Conclusion and summary
The neoclassical impurity density up/down asymmetry and radial particle transport in the elongated tokamak plasmas is investigated by applying the Miller equilibrium model. The θ-independent poloidal magnetic field is weakened by the plasma elongation with the plasma current fixed. Due to this the plasma elongation enhances the neoclassical impurity density up/down asymmetry by the factor (1+κ 2 ) 2 4κ . Accordingly the impurity Pfirsch-Schlüter neoclassical particle transport, which is proportional to the impurity density up/down asymmetry, is also magnified by the plasma elongation by the enhancement factor ( 1+κ 2 2κ ) 2 . Both the neoclassical up/down asymmetry of the impurity density and the impurity Pfirsch-Schlüter neoclassical particle transport are under-predicted by the neoclassical theory with the circular plasma cross section assumption. In addition, it will provide us with a possible method to control the impurity neoclassical particle transport through the plasma elongation control.