High-Z impurity neoclassical transport in the tokamak plasmas due to the single-null divertor configuration

A new mechanism for the high-Z impurity neoclassical particle transport in the tokamak plasmas due to the single-null divertor configuration is discovered for the first time. It will play an important role in the tokamak plasma pedestal region with the coexisting of the strong bulk ion radial gradients and the up/down asymmetry of the poloidal magnetic field. The outward (inward) high-Z impurity neoclassical particle transport will be driven with the B×∇B drift towards (away from) the X-point. The new finding indicates that the International Thermonuclear Experimental Reactor lower single-null divertor configuration will be beneficial for screening and flushing out the high-Z impurity in the pedestal region with the toroidal magnetic field in the normal direction ( B×∇B drift towards the X-point).


Introduction
Impurity content is inevitable due to the plasma-material interactions in the magnetic confinement fusion devices, such as tokamak and stellarator.Impurity transport in the tokamaks has been an issue of great concern for several decades [1,2].Inward transport of the impurity ions from the tokamak edge into the core plasma will cause the impurity core accumulation, which will degrade the plasma performance and even terminate the discharge due to fuel dilution and power loss from radiation.
The mental material such as tungsten is now widely used for the first wall and divertor plates in the currently operated tokamaks and also will be applied in the International Thermonuclear Experimental Reactor (ITER) tokamak [3].
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Understanding the physical mechanisms for the high-Z impurity transport in the tokamak plasmas is one of the key issues to be solved to realize the long pulse steady-state operation for the currently operated tokamaks and the future fusion reactors.The transport of the impurity ions in the tokamaks could be due to the classical/neoclassical collisions and the turbulence.The classical collisional transport usually can be neglected in the tokamaks.For the high-Z impurity ions, the neoclassical collisional transport, which is due to the tokamak configuration and will be aimed to be discussed in the following, could compete with or even dominate over the turbulent transport [2].The impurity neoclassical particle transport is driven by the radial density and temperature gradients of the bulk ions.The so-called 'temperature screening' on the impurities is due to the bulk ion temperature radial gradient [1,4], which has been verified in the tokamak experiments [5,6].Helander have extended the neoclassical transport theory to be applicable for the steep bulk ion radial gradients [7,8].The high-Z impurity neoclassical particle transport will be weakened with the bulk ion radial gradients strong enough [2,7,8].In this paper, a new high-Z impurity neoclassical particle transport mechanism in the tokamaks due to the the single-null divertor configuration will be discovered for the first time.
The divertor configuration in the tokamaks was originally designed to reduce the impurity content and improve the plasma confinement [9].A null or X-point in the poloidal magnetic field will be formed.Near the X-point the magnitude of the poloidal magnetic field is small.The ITER tokamak is designed to be operated in the lower single-null (LSN) divertor configuration [10].In the single-null diverted tokamaks the poloidal magnetic field will be up/down asymmetric in the edge plasma, which is usually neglected in the neoclassical transport theory.It will be pointed out that this up/down asymmetry of the poloidal magnetic field can cause the formation of up/down asymmetry of the high-Z impurity density.Accordingly an additional high-Z impurity neoclassical particle transport will be driven.This new mechanism will play an important role in the tokamak pedestal region with the coexisting of the strong bulk ion radial gradients and the up/down asymmetry of the poloidal magnetic field.
The remaining part of this paper is organized as follows.In section 2, the theoretical model will be presented.In section 3, the impurity neoclassical particle transport due to the singlenull divertor configuration will be numerically demonstrated.In section 4, the summary and conclusion will be given.

Theoretical model
The impurity neoclassical particle transport in the non-rotating tokamak plasma is strongly related with the impurity density poloidal asymmetric distribution [11], which is determined by the impurity parallel momentum equation, Here B = I(ψ )∇φ + ∇φ × ∇ψ is the magnetic field, where φ is the toroidal angle of the torus and ψ is the poloidal magnetic flux; p Z is the impurity thermal pressure and F Zi is the friction force between the impurity and the bulk ions.The parallel ion-impurity friction force F Zi|| = B • F Zi /B is in the following form, Here m i and n i are the mass and density of the bulk ion respectively; i /(n i e 4 ln Λ) is the ionion collision time, where Z and n Z are the charge number and density of the impurity ion respectively; K Z and u are both the flux function and related with the poloidal rotation of the impurity and bulk ions, respectively.The poloidal variation of the bulk ion density is neglected due to its much lighter mass and the much lower impurity content.The scale length of the bulk ion radial gradients is defined as , where the prime denotes d ψ , and k is the coefficient of the ion thermal friction force due to the ion temperature gradient and depends on the bulk ions collisionality regime [12].The pressure gradient term for the high-Z impurity ions can be neglected due to its large charge number.The effects of the impurity parallel viscosity and the poloidal electric field are neglected in equation (1).Including these effects will not qualitatively change the results in this paper.
The impurity density poloidal asymmetric distribution can be decomposed into n Z = n Z0 (1 + n Zs sin θ P + n Zc cos θ P ), where n Z0 is the impurity poloidal-independent density, n Zs and n Zc denote the impurity density up/down and in/out asymmetry respectively.θ P denotes the poloidal position on a certain magnetic flux surface and is determined by the rule of equal arc length; θ P = 0 corresponds to the middle plane on the low field side and θ P increases in the anti-clockwise direction on the magnetic flux surface.Then equation (1) will be where the dimensionless parameters . The flux-surface-averaged is defined as ⟨A⟩ = ¸Adθ P /(B • ∇θ P )/ ¸dθ P /(B • ∇θ P ).To obtain equation ( 3), the impurity ion temperature is thought to be equal to the bulk ion temperature due to the sufficient ionimpurity energy exchange and uniform in the magnetic flux surface due to the fast parallel heat transport within the magnetic flux surface.The parameter g plays an important role in the impurity neoclassical transport both in the conventional and extended neoclassical theory as has been shown in [7,8].It is strongly related with the impurity charge number and the radial gradients of the bulk ion density and temperature.It could be positive and negative, which depends on the bulk ion collisionality regime and the density and temperature radial gradients.In the conventional neoclassical transport theory, the radial gradient of the bulk ion temperature will cause the so-called 'temperature screening' effect with g < 0 and the radial gradient of the bulk ions density will cause the impurity core accumulation with g > 0. In [7,8], the ion neoclassical transport nonlinearly dependent on the bulk ion radial gradients was investigated through equation ( 3).While the up/down asymmetry of the poloidal magnetic field B P was not considered there.It is appropriate in the tokamak core plasmas.It will be shown that the up/down asymmetry of B P can drive the non-negligible impurity neoclassical particle transport in the single-null diverted tokamak plasma pedestal region.
The effect of the plasma elongation on the impurity density up/down asymmetry and neoclassical particle transport have been investigated with the Miller equilibrium model based on equation (1) in [13].Here it will be extended to investigate the impurity neoclassical particle transport in the singlenull diverted tokamak plasmas.By multiplying the factor 1, sin θ P , cos θ P respectively on both sides in equation ( 3) and then taking the flux-surface-averaged operation, the equations to determine the impurity density poloidal asymmetry can be obtained, where H 1 − H 5 and J 1 − J 5 are the flux-surface-averaged operations and defined as The integrations H 2 , H 4 , J 2 , J 4 are related with the up/down asymmetry of B P and go to zero without such asymmetry.Equation ( 4) can be approximately simplified to The coupling terms due to the impurity density poloidal asymmetries have been neglected in the first order equation.With these terms retained, the key results will not be changed qualitatively [13].By combining equations ( 5) and ( 6), the impurity density up/down will be derived with the help of equation (7), where In equation ( 8), the term proportional to α c is due to the in/out asymmetry of the toroidal magnetic field [7,8,13]; the term proportional to α s is driven by the up/down asymmetry of the poloidal magnetic field, which can be caused by the single-null divertor configuration in the tokamak edge.This new mechanism to induce the up/down asymmetry of the impurity density will cause an additional impurity neoclassical particle transport accordingly, which will be discussed in the following.
Actually the up/down and in/out asymmetries of the impurity density are coupled together through the ion-impurity parallel friction force.The nonlinear dependence of the impurity density up/down asymmetry on g proposed by Helander [7,8] and the effect of the B P up/down asymmetry on the formation of the impurity density up/down asymmetry proposed here are both due to this coupling.With g increasing, this coupling effect will become stronger and cannot be neglected.
The impurity neoclassical particle transport can be derived from the toroidal component of the impurity momentum equation, where ,σ B = I/ |I| represents the direction of the toroidal magnetic field B T and σ B = +1 denotes the toroidal magnetic field in the normal direction (perpendicular to the paper and pointed out).Actually Γ ψ Z0 is not sensitive to the up/down asymmetry of B P .The up/down asymmetry of B P affects the impurity neoclassical particle transport mainly through its influence on the impurity density up/down asymmetry formation as given by equation (8).Equation ( 1) has been used to obtain equation (9).
From equation ( 9), it can be obtained that the total impurity neoclassical particle flux is proportional to the impurity density up/down asymmetry as the conventional theory [7,8,13].The impurity density up/down asymmetry will disappear for the zero impurity neoclassical particle flux, which is consistent with the steady state numerical simulations for the impurity density profile in [14,15].The zero impurity density up/down asymmetry results from the zero g, which can be read from equation (8).When the impurity density radial gradient term is balanced with the bulk ion radial gradients term, g will be zero and the steady state pointed out in [14,15] will be reached.In this steady state, the up/down asymmetry of the poloidal magnetic field will not affect the impurity density profile, which will be determined by the zero g.But before reaching this steady state, the up/down asymmetry of the poloidal magnetic field will affect the impurity density profile through affecting the impurity neoclassical particle transport, which is the key point to be discussed in this paper.
The impurity neoclassical particle flux can also be written in the following form, The subscripts 'HL' and 'SN' in equation ( 10) denote the impurity neoclassical particle flux given by Helander's theory and driven due to the single-null divertor configuration respectively, Without the up/down asymmetry of the poloidal magnetic field, i.e. α s = 0 equation (10) will reduce to the Helander's theory [7,8].The driven impurity particle flux is scaled as g/(1 + 4λg 2 )and is weakened with |g| ≫ 1/ √ 4 |λ| ≈ 1.The direction of the driven impurity particle transport depends on the sign of σ B α c g, which is actually determined by the radial gradients of the bulk ion density and temperature and the collisionality regimes.With α s ̸ = 0 the impurity neoclassical particle flux driven by the single-null divertor configuration is also scaled as g 2 /(1 + 4λg 2 ) and saturates to σ B α s /4λ with |g| ≫ 1/ √ 4 |λ| ≈ 1.The direction of the driven impurity neoclassical particle transport depends on the sign of σ B α s , which is determined by the toroidal magnetic field direction and the up/down asymmetry of B P .This will be shown further in the following numerical demonstrations.

Numerical demonstrations
To demonstrate the impurity neoclassical particle transport due to the single-null divertor configuration, the analytical Solov'ev solutions for the diverted tokamak plasma equilibrium is employed.Follow the work of Cerfon and Freidberg [16], an upper and a lower single-null diverted tokamak plasma equilibrium are generated as shown in figures 1(a) and (b).The parameters for the modeled equilibrium are: the inverse aspect ratio ε = 0.3, the plasma elongation κ = 1.8, the plasma triangularity δ = 0.3, the X-point position X sep = R/R 0 = 0.8 and Y sep = Z/R 0 = ±0.54.The contour plots of the magnetic flux surfaces and the normalized B P magnitude are shown in figure 1.It is clearly shown that B P in the edge of the single-null diverted tokamak plasma is up/down asymmetric due to the formation of the X-point.
With the tokamak plasma equilibrium given in figure 1, the impurity neoclassical particle transport due to the single-null divertor configuration can be investigated.The toroidal magnetic field is set in the normal direction.γ is not sensitive to the variation of g and will be set to zero as done in [7,8] to seek out the main physics proposed here.The radial dependence of α s due to the up/down asymmetry of B P in the single-null divertor configuration is shown in figure 2. α s approaches to zero toward the plasma core and increases to its maximum value toward the plasma edge.The α s effect will be magnified with the steeper bulk ion radial gradients in the tokamak pedestal region.The dependences of the impurity neoclassical particle flux due to the single-null divertor configuration on g on the normalized magnetic flux surface ψ N = 0.8 and ψ N = 0.95 in the tokamak edge are shown in figure 3. It can been seen that there is little difference on these two flux surfaces.The existing of the single-null divertor configuration effect on the impurity ions neoclassical particle transport is determined by the combined effect of the non-zero α s and the large |g| magnitude.
The LSN divertor configuration drives the outward impurity particle transport, while the upper single-null divertor configuration drives the inward impurity particle transport.This is due to the B P up/down asymmetry is reversed as the divertor configuration changes from the LSN to the upper single-null.If the toroidal magnetic field is reversed to the anti-normal direction, the situation will be reversed correspondingly.The driven impurity particle flux increases with |g| increasing and saturates as |g| large enough, which have been pointed out above.The impurity neoclassical particle transport predicted  by the Helander's theory is also shown in figure 3. The impurity neoclassical particle transport due to the single-null divertor configuration is comparable and even larger than that predicted by the Helander's theory with |g| large enough.
The total impurity neoclassical particle flux for the lower and upper single-null cases are also shown in figure 3. The single-null divertor configuration starts to affect the total impurity neoclassical particle transport with |g| approximately larger than 1.With g < 1 it is in the conventional impurity neoclassical transport theory region.With g > 1 it is in the extended impurity neoclassical transport theory region, in which the impurity neoclassical particle transport will be affected greatly by the strong bulk ion radial gradients pointed out in [7,8] and the single-null divertor configuration proposed here.
As g < 0, the LSN configuration enhances the outward impurity particle transport while the upper single-null configuration reduces the outward impurity particle transport and even reverses the impurity particle transport to the inward direction.As g > 0, the upper single-null configuration enhances the inward impurity particle transport while the LSN configuration reduces the inward impurity particle transport and even The impurity neoclassical particle flux due to the single-null divertor configuration versus the bulk ion radial gradients parameter g.The impurity neoclassical particle flux has been normalized to Γ ψ Z0 .The positive (negative) value denotes the outward (inward) particle transport.The labels 'LSN', 'USN' and 'HL' denote the impurity neoclassical particle flux due to the lower and upper single-null divertor configuration and predicted by the Helander's theory respectively.The labels 'LSN_T' and 'USN_T' denote the total impurity neoclassical particle flux with the lower and upper single-null divertor configuration respectively.
reverses the impurity particle transport to the outward direction.The impurity neoclassical particle transport due to the single-null divertor configuration is non-negligible with |g| large enough and it should be included in the total impurity neoclassical particle transport calculation in the tokamak edge plasma.
It is found that the B × ∇B drift is towards (away from) the X-point if the toroidal magnetic field is in the (anti-) normal direction.Actually the direction of the driven impurity neoclassical particle transport due to the single-null divertor configuration can be summarized more generally as the following: if the B × ∇B drift is towards (away from) the X-point, the outward (inward) impurity neoclassical particle transport will be driven.This will be helpful to judge the direction of the driven impurity neoclassical particle transport conveniently.
The dimensionless parameter g is proportional to Z 2 and the bulk ion radial gradients.The dependence of |g| on R/L nT for the different impurity charge number are shown in figure 4.Here L −1 nT = |d ln n i /dr + kd ln T i /dr|, q = εB T /B P , ε = r/R 0 , r is the minor radius at the middle plane in the low field side and R 0 is the major radius at the magnetic axis.For the low-Z impurity |g| is much smaller than 1 even if the bulk ion radial gradients are strong.While for the high-Z impurity |g| will be larger than 1 with the bulk ion radial gradients strong enough.Therefore the new mechanism due to the single-null divertor configuration for the impurity neoclassical particle transport proposed here will only have remarkable impart on the high-Z impurity such as the tungsten impurity and have negligible effect on the low-Z impurity.For a certain high-Z impurity, there is a critical bulk ion radial gradients for the single-null divertor configuration starting to play an important role in the neoclassical particle transport.It will happen in the single-null diverted tokamak plasma pedestal region, where the up/down asymmetry of the poloidal magnetic field and the strong bulk ion radial gradients coexist.g also depends on the bulk ion collisionality, which is related to the bulk ions density and temperature.The contour plot of |g| value for the high-Z impurity ions with Z = 40 in the n i − T i space for R/L nT = 10.0 and R/L nT = 20.0, which can be achieved in the pedestal region in both the currently operated tokamaks and the future fusion reactors, is shown in figure 5.Both in the lower (left bottom region in figures 5(a) and (b)) and higher (right top region in figures 5(a) and (b)) density-temperature operated regions, |g| could be larger than 1 for the high-Z impurity ions.This indicates that the singlenull divertor configuration could play an important role in the high-Z impurity neoclassical particle transport in the pedestal region for both the currently operated tokamaks and the future fusion reactors.Here only the neoclassical convective particle transport for the high-Z impurity ions is discussed.The neoclassical and turbulent diffusion particle transport, the turbulent convective particle transport and the impurity source should be included to model the high-Z impurity density profile evolution.The impurity source is usually due to the plasma-material interactions in the tokamak edge.The screening of the impurity ions in the tokamak edge is an important aspect to prevent the impurity core accumulation.The transport mechanism for the high-Z impurity ions proposed in this paper will be helpful to deeply understand how the impurity ions enter or are flushed out through the pedestal region.
Controlling the high-Z impurity content is essential for the ITER tokamak to realize the high plasma performance and steady-state operation.The new finding in this paper indicates that the ITER operation with the LSN divertor configuration and the toroidal magnetic field in the normal direction is favorable for the high-Z impurity expulsion in the pedestal region.It will be beneficial for preventing the high-Z impurity into the core plasma and improving the plasma performance.While it should be pointed out here that the high-Z impurity core accumulation is related with the other important factors, such as the divertor closure and the impurity leakage and transport in the SOL.Also it is essential to consider the variations in the density and distribution of high-Z impurity source between the normal and anti-normal toroidal magnetic field directions.

Summary and conclusion
The high-Z impurity neoclassical particle transport in the tokamak edge plasmas due to the up/down asymmetry of the poloidal magnetic field in the single-null divertor configuration is proposed.This new transport mechanism becomes more important with the bulk ion radial gradients increasing.The inward or outward high-Z impurity particle transport can be driven.It depends on whether the direction of B × ∇B drift towards or away from the X-point.This additional contribution to the high-Z impurity particle transport should be included in the calculation of the total particle transport in the tokamak plasma pedestal region.For the ITER tokamak operation with the toroidal magnetic field in the normal direction will be beneficial for screening the high-Z impurity in the pedestal region and reducing the high-Z impurity content in the core plasmas.It will play an important role in improving the ITER plasma performance.

Figure 1 .
Figure 1.The contour plots of the magnetic flux surfaces and the normalized B P magnitude in (a) a lower and (b) a upper single-null divertor tokamak plasma.

Figure 2 .
Figure 2. The radial dependence of αs for the lower and upper single-null divertor configuration.

Figure 3 .
Figure3.The impurity neoclassical particle flux due to the single-null divertor configuration versus the bulk ion radial gradients parameter g.The impurity neoclassical particle flux has been normalized to Γ ψ Z0 .The positive (negative) value denotes the outward (inward) particle transport.The labels 'LSN', 'USN' and 'HL' denote the impurity neoclassical particle flux due to the lower and upper single-null divertor configuration and predicted by the Helander's theory respectively.The labels 'LSN_T' and 'USN_T' denote the total impurity neoclassical particle flux with the lower and upper single-null divertor configuration respectively.