1.Introduction

The Chinese Fusion Engineering Test Reactor(CFETR),a next-generation full superconducting fusion reactor,is being developed in China.The conceptual design has already been fulfilled.Recently,the CFETR has been redesigned to a larger size to make it easier to transfer from Phase I to Phase II[1].The major plasma parameters of CFETR in Phase I and Phase II are listed in table 1.Phase I has 200 MW fusion power incorporating 7.5 MA plasma current,and phase II has 1GW fusion power incorporating 10 MA plasma current.

The standard single null divertor(SND)is not suitable for CFTER since the heat flux is expected to outstrip the currently allowed 10 MW m-2steady-state limit[2].For the previous,smaller,CFETR,quasi-snow flake-divertor(QSFD)con figurations were first established to solve this problem[3].On that basis,three types of snow flake(SF)con figuration were attained[4].The SF con figuration has a second-order null point,i.e.the first derivatives of the magnetic field also vanish at the null point,and the poloidal plane is split into six sectors by the separatrix[5].The dimensionless distance(σ=　D/ a)parameterizes the proximity as an ideal SFD by setting ‘D’as the X-point separation and ‘a’as the plasma minor radius.This con figuration,called exact SF(exhibited in figure 1(a)),refers to a particular mathematical solution never obtained during experiments.It makes sense only when being discussed with regard to SF parameters.There are two comparatively stable con figurations,namely snow flake plus(SF-+),and snow flake minus(SF-).As presented in figure 1(b),SF+has on two first-order null points,both X-points carry an associated separatrix;the ‘X’de fines the last closed- flux surface(LCFS)and is termed primary,while the‘A’is termed secondary.The secondary X-point introduces two additional strike points(SPs)to the SF.The SP with field lines closest to the primary separatrix is called the primary SP,and the others are termed secondary SPs.For SF+,‘A’constantly pertains to the private- flux region and the scrape-off layer(SOL) field lines are merely connected to primary SPs.In contrast,the‘A’in SF–pertains to the SOL.The SF– con figuration can be classified as SF–with a high- field side(HFS)and SF–with alow- field side(LFS),depending on the radial location of‘A’in the SOL.The SF–con figurations in this paper generally pertain to HFS SF–.In figure 1,the SPs are numbered in a counter-clockwise direction.In particular,for SF–two primary SPs(SP3,SP4)are on the low- field side,and SP1 and SP2 pertain to secondary SPs.For the flux surfaces lying between the primary separatrix and the secondary separatrix,all field lines on the HFS go to SP3.The field lines of the surfaces with between the secondary separatrix on the LHS go to SP1.SP1,SP3,SP4 are all connected to the SOL,and SP2is magnetically disconnected from the other SPs.In[4]the SF con figurations were studied and the poloidal field(PF)coil system was evaluated.The results indicate a marked increase in connection length and flux expansion.About 30 Wb(～300s)can be supplied to the flattop phase of the 10 MA inductive scenario of SFD con figurations with the PF coil system.

Table 1.Performance of CFETR plasma in Phase I(cases A and B)and Phase II(case C).

Figure 1.Geometries of three types of SF con figuration in the vicinity of the nulls:(a)exact SF;(b)SF+;(c)SF–.The red lines pass through the primary X-points;The blue lines pass through the secondary X-points.In(c),the blue magnetic surface is 1.23 mm away from the LCFS at the outer midplane and four strike points are plotted.

Table 2.PF coil parameters.ΔR represents the radial width of the coils and ΔZ denotes the vertical height.The coil current limit is 50 kA/turn.

Coil　R(m)　Z(m)　　ΔR(m)　　ΔZ(m)　Turns CS1U　1.85　1.025　1　2.05　738 CS2U　1.85　3.075　1　2.05　738 CS3U　1.85　5.125　1　2.05　738 CS4U　1.85　7.175　1　2.05　738 CS4L　1.85　-7.18　1　2.05　738 CS3L　1.85　-5.13　1　2.05　738 CS2L　1.85　-3.08　1　2.05　738 CS1L　1.85　-1.03　1　2.05　738 PF1U　4.32　9.834　1.1　1.5　448 PF2U　12.826　8.004　1.1　1.1　225 PF3U　15.445　3.15　1.1　1.1　225 PF1L　4.32　-9.83　1.1　1.5　448 PF2L　12.826　-8　1.1　1.1　225 PF3L　15.445　-3.15　1.1　1.1　225 DC1　6.824　-10　1.1　1.1　225 CD2　9.924　-9.67　1.1　1.1　225

CFETR has been updated to a larger size(R = 6.6 m,a = 1.8 m,BT = 6–7T),requiring the new PF coil system to be re-evaluated.The position,size and turns of the PF coils are presented in table 2.The novel superconducting PF coil system comprises eight central-solenoid(CS)coils,sixPF coils and two additional divertor coils(DC),enabling CFETR to generate several con figurations of an advanced divertor.This system drives the inductive current and controls the feedback of plasma position.The limits of these PF coils,inclusive of coil current limits,coil field limits etc,must,overall,be maintained.This paper lays particular stress on the coil current limits.By adopting the high-temperature superconductor Bi-2212,CFETR will have a maximum magnetic flux of～240Wb in the plasma area[1],and a poloidal flux in excess of 450 Wb will be employed for the entire discharge.To accurately ascertain the capability of the PF coil system,a batch of plasma equilibria was established for Phases I and II(encompassing SND and SF con figurations),and the coil currents in the flattop phase were acquired for a range of internal self-inductances li3 and flux states.As presented in[6],li3 and flux state are both key parameters that determine the coil current.The flux state is defined as the poloidal magnetic flux from all coils linked over the entire plasma.li3 is defined as[7]

where V is the plasma volume,BPis the poloidal magnetic field and R0is the primary radius.

To quickly and readily acquire the currents in the PF coils in the presence of a discharge,static equilibrium calculations were conducted using the TEQ module[8]at some fiducial points of the flattop for a range of flux states.This‘static time-slice equilibrium’is dependent on the flux state.The flux state and the time in a discharge are exclusively related to each other.The difference between two flux states is equivalent to the volt-second consumption between two corresponding moments in the case of a discharge.The overall volt-second consumption in the current ramp-up phase is defined as[9]:

The Phase I plasma has Ip = 7.5MA,magnetic field BT = 6.0 T,major radius R = 6.6 m,minor radius a = 1.8 m and normalized toroidal beta βN = 1.6.The plasma equilibrium is acquired by solving the Grad–Shafranov equation regarding typical H-mode plasma current and pressure pro files.

In this subsection we shall discuss the flexibility of the PF coil system in CFETR and whether the SND and SF con figurations can be sustained over the whole discharge period while the coil currents are within their limits.Obviously,these con figurations can be sustained during the whole of an adequately non-inductive(or steady-state)discharge.With regard to an inductive discharge,the question is more complicated.The equilibria were acquired at some fiducial points( flux states)on the flattop burn phase,and on that basis the values of coil current were attained.A batch of current density and pressure pro files was adopted to establish a series of equilibria incorporating different li3.The PF coil currents of three plasma con figurations were acquired on the flattop phase with li3 ranging from 0.6 to 1.0,as well as the flux states.Figure 6 shows a flux state versus li3 operating space diagram,showing the region where PF coil currents are within their limits.In the 7.5 MA inductive scenario,the PF coil system can provide a length of flattop(LOF)for the SND in excess of 300 Wb,and the flux states are con fined by accessing the higher li.Even though the flux states of SF+and SF–are also indicated to be substantially con fined by accessing the lower li,the PF coil system can at least offer in excess of 125 Wb and 100 Wb to the flattop phase,respectively.In line with a scale of plasma resistance,the flattop time can be approximately estimated[18].In figure 6 the maximum of the LOF lasts of the order of seconds.

where ΔΨbreakdowndenotes the flux consumed in the course of generating the loop voltage and magnetic- field null point for breakdown,Ψextrefers to the external poloidal flux requirement and ΔΨresand ΔΨindrepresent the resistive and inductive flux consumptions,respectively.Starting from the null field(～210 Wb),a batch of fiducial points in the discharge are defined,inclusive of the start of discharge(SOD),point in ramp(PIR),start of flattop(SOF),middle of flattop(MOF)and end of flattop(EOF)[6].For instance,hypothesizing that li3 is settled at 0.8,and the overall flux consumption in the ramp-up phase is 122.3 Wb(from formula(2)).For this reason,the flux state of the SOF with li3 = 0.8 reaches 77 Wb.

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2.SF con figuration of 7.5 MA Phase I plasma

where

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The Phase II plasma is characterized byIP　=　10MA,BT　=　6.0T,major radiusR=6.6m,minor radiusa=1.8m and normalized toroidal beta　β　N=　2.6.The plasma current shows substantial growth,resulting in a large increment of the coil currents in the PF coils,especially for SF–.Unfortunately,the PF system cannot sustain the 10 MA SF–configuration,as the DC coil currents for 7.5 MA SF–are extremely close to their limit(50 kA/turn)and exceeded their limit at the SOF in the 10 MA scenario.As shown in figure 7,the values of LCandfexpfor the 10 MA equilibria are similar to those of the 7.5 MA equilibria:fexpand the LCof SF+are about 1.5 times larger and 1.8 times longer,respectively,than those of the SND.Figure 8 shows that the PF coil system can provide 270–303Wb(5260–5900 s)for the flattop of the 10 MA SND and 175–207Wb(3410–4030s)for that of SF+.

2.1.Flux expansion and connection length

where BTis the toroidal magnetic field.In the vicinity of the null points,the toroidal field|BT|evidently outstripsIn this regard,the integral can be denoted asLC=A small∣BP∣shall accordingly give rise to a marked increase in LC.This indicates that the length of the magnetic- field line in the SOL approaching the null points predominantly contribute to LC.That the second null point reduces in the divertor separatrix region is illustrated in figure 4.Figure 5(a)indicates the radial pro files of LCfor various divertor con figurations from the outer MP to the LFS target.Withinξ＜2mm,the LCof SF+increases approximately 1.5–1.9 times,and the value for SF– increases 1.2–1.7 times.In particular,the bene fits of the SF–geometric structure are subtle and non-obvious.As mentioned in section 1,all field lines between the primary separatrix and the secondary separatrix(0　＜　ξ　＜　1.23mm)on the HFS move towards SP3.For　ξ＞　1.23mm,those field lines on the LFS approach SP1.The integration of the total LCfrom the LHS to the HFS is non-contiguous from the secondary separatrix.Figure 5(b)elucidates the non-contiguous overall LCof SF–.The LCof SF–within0　＜　ξ　＜　1.23mm is non-monotonic,as the surfaces far from point X approach point A.

The impact exerted by low BPcan also extend the connection length,LC.LCis by and large taken as the line integral along the magnetic- field lines in the SOL from the outer MP to the outer divertor target in the LFS.It can be denoted as

The second-order null modi fies the geometry of the SF and reduces the gradient of BPapproaching the null[11].The value of B2Pin the vicinity of the second-order null rises as distance ras opposed to r in the SND con figuration[12].Accordingly,the distance between flux surfaces in the SOL approaching the null point increases.This effect is termed‘ flux expansion’,fe　xp,and it affects the thickness of the SOL and may consequently increase the radiating volume.For quantitative analysis of the impact offe　xp,two geometrical parameters are defined[13]:Δ,the minimum distance between the X-point and the flux surface on the low- field side,andξ,the distance between the flux surface and the primary separatrix at the outer mid plane(MP).In figure 3,the value offor the SF con figurations and for a SND is plotted as a function ofξ.Forξ　＜ 2mm,SF+and SF–have values over 1.5 times larger than that of the SND.These results are similar to those found in the TCV SF experiments[13].

Figure 2.7.5 MA plasma divertor con figurations with flux state　Ψ =-　100 Wb and PF coil currents(kA):(a)SND;(b)SF+;(c)SF–.The structures of the toroidal field(TF)coils,vacuum vessel, first wall and divertor are also shown.

Figure 3.Flux expansion(Δ/ξ)which is calculated near the X-point on the LFS for a SND(black),a SF+(red),a SF–(blue).

Figure 4.B　P/BTof the SND and SFD con figurations,as a function of connection length LCfrom the outer MP to the outer divertor target.

Figure 5.Two definitions of LC.(a)LCfrom the MP to the outer divertor plate.(b)The total LCfrom SP4(or SP2 in the case of a SND)to the other divertor target.The red dashed line denotes λq ～ 2.7 mm.The green dashed line indicates the distance(～1.23 mm)at the outer MP between the surfaces which pass through point A and the LCFS in the SF–con figuration.

Figure 6.Flux state versus li3 diagram indicating the LOF for the plasma con figurations with Ip = 7.5 MA.The solid lines show where the coil’s current limit is exceeded.The colored dashed lines display the locations of the SOF of the SND(black)and the SF(green)within the li3 range 0.6–1.0.The maximum LOF(in Wb and s)for each con figuration is shown by the dashed arrows.

The power decay length of 7.5 MA CFETR plasma,can be estimated from the scaling law,[14],showing that the heat flux in the SOL should primarily reach SP3,SP4 and SP1 in the SF– con figuration.λqdepends on the balance of parallel and cross- field transport over the SOL.A longer LCmeans a longer residence time of a particle in the SOL,and should stimulate cross- field transport resulting in a widerλqin a common SOL and greater radiative losses in the private- flux region.In[15],for SF–,analysis of the experiments showed that the power of one of the SPs is divided to two SPs with dominant parallel transport,and in addition potentially by cross- field transport to the fourth SP.For SF+,the power exhaust in the common flux region can be redistributed to the private- flux region under the impact of cross- field transport,and much larger power fluxes to the secondary SP3 are triggered by a radial transport channel[16].The results of experiments in[17]indicate that during edge-localized modes,the fraction of the exhaust power reaching the additional SPs outstrips that in L-mode.

2.2.The length of the flattop in the 7.5 MA inductive scenario

and

3.Results for 10 MA Phase II plasma

自动盖章环节结束以后进入自动下料，机构上抬至中间上方位置，然后继续移动至右侧上方位置，机构下放放纸后回到上料初始位置。子程序如图6。

For the SFD con figuration,coils CS3L,CS4L,DC1,DC2 and PF1L are conducive to generating two null points and controlling the locations of the nulls by changing their coil currents.The SF con figuration can also be used with the H-mode[10].A SF+con figuration with two first-order null points(X,A)andσ=0.5is generated from an H-mode SND equilibrium.On that basis,the HFS SF–con figuration is also established by moving point A to the SOL on the HFS.These three plasma con figurations(Ψ = -100 Wb,li3 = 0.8)and their currents in the PF coils are shown in figure 2.

Figure 7.(a)Flux expansion(Δ/ξ)which is calculated near the X-point on the LFS for a 10 MA SND(black)and a 10 MA SF+.(b)LC from the MP to the outer divertor plate.

Figure 8.Flux state versus li3 diagram showing the LOF for the 10 MA plasma con figurations.The solid lines indicate where the coil current limit is exceeded.The green dashed line shows the location of the SOF for the li3 range 0.6–1.0.The maximum of the LOF(in Wb and s)for each con figuration is plotted by the dashed arrows.

4.Conclusion

In this paper,the 7.5MA SF geometries(SF+,SF–)and 10 MA SF+plasma con figuration were obtained with a new design for the PF coil system,and their properties were analyzed by TEQ code.The SF con figurations have lower BP values in the vicinities of the null points,resulting in larger flux expansion and a longer connection length.fexpof the 7.5MA SF+and SF–are over 1.5 times larger than that of the SND.The LCof the 7.5 MA SF+is increased about 1.5–1.9 times,and the value of the 7.5 MA SF– shows a 1.2–1.7 times increment.The results for the 10 MA SF+con figuration are similar to those for the 7.5 MA SF con figurations.fexpand LCof SF+are about 1.5 times larger and 1.8 times longer,respectively,than those of the SND.

Compared with the smaller CFETR,the LOF of the SND and the SF con figurations are extended notably[4].In the 7.5MA inductive scenario,the PF coil system can support a flattop phase to the SND of over 300 Wb(1660 s).For SF+,it has a LOF of 125–278 Wb(690–1540 s),and the value for SF– is 100–218 Wb(550–1210 s).In the 10 MA inductive scenario,the PF coil system can provide 270–303 Wb(5260–5900 s)for the flattop phase of the SND and 175–207 Wb(3410–4030 s)for that of SF+.These results indicate that CFETR has extensive capability for long-pulse operation in 7.5 and 10 MA inductive scenarios.This paper is still a preliminary work.An advanced magnetic con figuration divertor has been designed for CFETR.More simulations and analysis will be carried out in the future.

This work was supported by the National Magnetic Confinement Fusion Program under grant nos 2014GB106001,2014GB110003 and 2013GB111000,and by National Natural Science Foundation of China under grant no.11675221.

“当你在花园里喂鸟时，试着在喂食器附近搭一根树枝，这样鸟儿在进食之前会在树枝上栖息一会儿。”职业鸟类学家David Tipling说到。

其中，X={X1，X2，…Xp}为包含p个解释变量的向量，β={β0，β1，β2，…βp}为模型参数，可通过最大似然函数估计得到。很显然，液化概率PL(X)在0和1之间。记Yi=1和Yi=0分别为液化和非液化的情况，则似然函数L(β)可表示为：

References

[1]Wan Y X et al 2017 Nucl.Fusion 57 102009

[2]Mao S F et al 2015 J.Nucl.Mater.463 1233

[3]Luo Z P et al 2014 IEEE Trans.Plasma Sci.42 1021

[4]Li H et al 2017 Fusion Eng.Des.121 117

[5]Ryutov D D 2007 Phys.Plasmas 14 064502

[6]Kessel C E et al 2009 Nucl.Fusion 49 085034

[7]Elahi A S and Ghoranneviss M 2010 J.Fusion Energy 29 76

[8]Samaddar D et al 2013 J.Phys.:Conf.Ser.410 012032

[9]Nassi M 1993 Fusion Technol.24 50

[10]Piras F et al 2010 Phys.Rev.Lett.105 155003

[11]Ryutov D D and Soukhanovskii V A 2015 Phys.Plasmas 22 110901

[12]Soukhanovskii V A et al 2011 J.Nucl.Mater.415 S365

[13]Piras F et al 2009 Plasma Phys.Controlled Fusion 51 055009

[14]Eich T et al 2011 Phys.Rev.Lett.107 215001

[15]Reimerdes H et al 2013 Plasma Phys.Controlled Fusion 55 124027

[16]Lunt T et al 2014 Plasma Phys.Controlled Fusion 56 035009

[17]Vijvers W A J et al 2014 Nucl.Fusion 54 023009

[18]Wesson J 1989 Phys.Today 42 78