1.Introduction

The Solovev equilibria[1],supplemented by the vacuum solution by Zheng et al[2]have been used extensively for the systematic study of the theoretical properties of an axisymmetric plasma confinement device for its design,operation and/or modification.The task starts with a library of plasma equilibria that covers the target operation space of the device.An efficient and accurate method of generating a consistent set of plasma equilibria is extremely desirable.This is also needed when we want to test the validity of new computation tools.For an axisymmetric confinement device such as CFETR[3,4]and KTX[5],this means an efficient and accurate way to finding solutions of the Grad–Shafranov(GS)equation.The GS equation is a second order nonlinear partial differential equation(NLPDE)of the poloidal flux function ψ with multiple source terms due to the toroidal current source Jφ.A NLPDE is usually not easy to solve.If the problem can be simplified to be a linear partial differential equation(LPDE),and also if the solutions due the different source terms can be made independent from each other,then the solutions due to each source terms are superposable.The situation is then much improved.The key is therefore in adopting a simplified current pro file.On the other hand,the important parameters that characterize a plasma equilibrium include a set of integrated parameters and a set of intrinsic parameters.The integrated quantities include the total toroidal current,total pressure,total toroidal flux,over-all size,etc.The intrinsic quantities consist of a set of dimensionless parameters such as β(the ratio of total plasma pressure to total magnetic pressure),li(the internal inductance of the plasma),and a set of shaping parameters,such as elongation,triangularity,squareness,indentation of the boundary shape of the plasma cross-section.Thus,we can rephrase the question as what is the maximum capability of the set of solutions of the GS-equation with simplified current pro files in representing the plasma of our device under study and what is the limitation?

The first major contribution in tackling this problem was made by Solovev[1].He parametrized the current source in terms of simple(constant)parameters and expressed the solutions in terms of a set of polynomial functions in{R, Z}.A more important aspect of his approach is that he first assumed the plasma covered the whole space,without specifying its boundary.After the global solution was found,he then specified the(closed)boundaryψb.Because the GS equation is an elliptic PDE,according to the virtual casing principle[6],once the boundary is chosen,we can always find a rearrangement of the external coil currents to match the distribution of magnetic field at the plasma boundary so that the full solution will be the same as that found by the GS equation on the inside and that of the vacuum solution on the outside.Therefore in his approach,Solovev made the following major simplification of the problem:(1)The current pro file inside the plasma is represented by constant parameters.(2)All the solutions outside of the selected boundary are ignored(or not expected to be correct).For many occasions,we are indeed concerned only with obtaining an exact solution of the GS problem inside the plasma boundary.We will denote all equilibrium solutions obtainable in this way as the Solovev equilibria,and use{Solovev}to represent them.The ψ in{Solovev}is well behaved and diverges to infinity as(R,Z)tends to infinity.No transcendental functions are invoked.This fact made{Solovev}also the simplest possible solutions.Solovev also analyzed this set of functions and showed that{Solovev}can indeed give a rich variety of equilibria with many interesting shapes.{Solovev}has been found to be extremely useful and is usually the first set of equilibria used for studying new con figurations or new theoretical plasma tools such as the stability codes.However,for researchers with experience in this field,it is well known that{Solovev}does not have complete shaping capability.A lot of the interesting shapes which should exist are not found in{Solovev}.Examples of these deficiencies are the limitations in the achievable elongation,triangularity,etc.

A second major advance in this direction was given by the work of Zheng,Wootton and Solano(ZWS)[2].They pointed out that for the current pro files given by Solovev,there was another set of possible solutions.This set is denoted here as{ZWS}.Because ZWS was solving the same problem as Solovev,therefore the additional current source in{ZWS}was zero.Members of the{ZWS}were additional vacuum solutions.Although there was no current source(at finite locations),the solution was not zero everywhere.These solutions represented ψ resulting from sources outside of our region of interest.They all diverged as the(R,Z)tends to infinity.Another common feature of members of{ZWS}was that they have a current singularity at the central φ axis.Although singular current values were not allowed inside of the plasma of interest,they should not be excluded if we were concerned only with the region inside of　ψ=ψ　b and this region does not include the central φ axis.With the inclusion of{ZWS},the combined set{Solovev-ZWS)was deemed to have complete shaping-capability.Using this set,we should come across no limitation in finding any intended plasma shapes.For instance,we should be able to reach any elongation,triangularity or some other plasma shape parameters as first pointed out by Cerfon and Freiberg[7].They utilized{Solovev-ZWS]and constructed various kinds of commonly encountered plasma equilibria and named the completeshaping capability as the ‘One Size Fits All’property of the{Solovev-ZWS}.We mention here that this complete-shaping capability is really tied to the topology of the torus and the property that all the curvature vectors of the plasma boundary surface point inwardly.

式中，Mjk是零阶矩（6个参数）标准矩张量，表示瞬时震源机制。其他符号是二阶矩：时间矩心Δτ，空间矩心Δξl，时间—空间矩Δ（τξl），双时间矩 Δ（τ2），和双空间矩 Δ（ξlξm）。（x，ξ，t，τ）为响应函数（RF），（x，ξ，t，τ）是响应函数的时间导数，（x，ξ，t，τ）是响应函数的空间导数，（x，ξ，t，τ）是二阶时间导数，（x，ξ，t，τ）是时间和空间导数的组合，（x，ξ，t，τ）是二阶空间导数。二阶矩有以下物理解释：时间矩心、空间矩心、破裂扩展向量、震源过程的持续时间和震源的几何特征（详见Adamováandílen，2010）。

Plasma flow is one intrinsic property of the plasma not considered in the original work by Solovev[1].Therefore our motivation is to consider the effect of a general plasma flow.The effect of flow on plasma behavior has attracted considerable interest in fusion plasma research.Plasma rotation has been observed in many tokamaks,particularly in neutral beam heating experiments[8].The plasma rotation could be driven by neutral beam injection or rf heating or could also result from spontaneous or intrinsic processes in the tokamak.In general,due to rotation,either toroidally or poloidally,both the energy transport and the macroscopic stability could be significantly affected.There were also many investigations on the effect of plasma flow and flow shear on the macroscopic plasma instabilities such as the internal[9,10]and external[11,12]kinks,the ballooning modes[13],and the resistive wall mode RWM[14,15].It was found that,in general a moderate flow shear is always stabilizing,while the bulk toroidal rotation can be either stabilizing or destabilizing.It is also well known that the shape of the magnetic surface has important effects on the stability of plasma.Especially,in the H-mode scenario it is found that a proper shaping of the plasma can stabilize the peeling mode near the separatrix substantially.In recent years there were some numerical and analytical work on these effects.However most of these studies dealt only with a plasma without flow.

The generalized Grad–Shafranov(GGS)equation of a plasma with a general rotation was first given concisely by Hameiri[16].Lately,Tasso et al[17]have investigated a tokamak plasma with incompressible flows and found that in such cases the GGS can be solved analytically with some simplifying　assumptions.Guazzotto　etalinvestigated numerically the effects of arbitrary flows on the equilibrium of tokamak plasmas by using the equilibrium code FLOW[18].In addition,current reversal equilibrium with flow[19]were also investigated numerically.The important work by Maschke and Perrin[20]considered the case of a toroidal plasma with a pure toroidal rotation,and show that it is still possible to obtain particular exact analytic solutions of the equilibrium with toroidal flow when particular pro files of the poloidal magnetic flux function and pressure are adopted.However,their work is restricted to the discussions of toroidicity and ellipticity only.The GGS equation for plasmas with a toroidal flow is also solved by Poulipoulis et al[21]for plasmas with a rectangular plasma boundary and with linear current pro file.They found solutions consisting of Bessel functions and that plasma flow could lead to a change in plasma topology.However,it is not clear whether these solutions can be generalized easily to other plasma shapes for the purpose of shaping studies.

We let

The rest of the paper is organized as follows:in section 2,we first present the formulation of the equilibrium for a tokamak plasma with a general flow.This results in a NLPDE for the poloidal flux function ψ with nonlinear source functions that are defined through a set of nonlinear constraints.The nonlinear constraints contain the effects of plasma flow.We examined the compatibility of the simplification of the problem with respect to the behavior of the constraint functions.For the case without flow,the problem is reduced to that originally studied by Solovev[1].We first show that with the addition of toroidal flow,the only meaningful generalization is to the case of a plasma with a pure toroidal flow at a constant Mach number.Next we showed that for a plasma with a poloidal flow,the only rather trivial generalization is to a plasma with zero density.Following Solovev,we further move the solution boundary to cover the whole poloidal plane.In addition we impose the internal boundary condition at the location of the magnetic axis.The result is an LPDE with boundary conditions imposed at the magnetic axis.The LPDE is then solved analytically.Utilizing the principle of superposition,the solution is decomposed into its component parts resulting from different source functions located at different locations.We discuss the properties of these solutions and their relationships to the intrinsic parameters.Details of this analytic study are given in section 3.In section 4,we discuss the asymptotic behavior of the solutions at the boundary and their appropriateness to be included in the solution space and tie this to the topology of the target plasma.It is followed by a discussion of the incomplete-shaping-capability of the set{SOLOVEV_ZWSm}for con figurations with curvature vectors at the boundary that point from the plasma outward(including the doublets and bean shaped plasmas).A summary is provided in section 5.The derivation of the vacuum solutions is given in appendix A.The relation of a part of these vacuum solutions to the usual solutions of the Poisson equation is discussed in appendix B.

2.Formulation of the axisymmetric equilibria for a plasma with flow and its simplification to a linear problem with linear superposable solutions

2.1.Formulation of the general equilibrium problem for axisymmetric plasma with flow

For the axisymmetric plasma,the magnetic field satisfies the divergence free condition and can be written as

ThusF=RBφ.The current density can be written as is the poloidal magnetic flux,andSimilarly,the equation of continuity demands that the momentum density can be expressed asThe Ohm’s law demands thatThe second part of this relation follows because from the first part we obtain0.Therefore the electrostatic potential　Φ=Φ(ψ)is a function of the poloidal flux.The third part of this equation is the standard definition of Ω.Substitution of equation(1)and the expressions forinto therelationship resulting from the Ohm’s law then gives uscomponent of this relationship demands thatV　=　V(ψ);and thecomponent gives us the relationshipSubstitution of this relationship into the expression of momentum density allows us to express the momentum density　asTaking　the divergence of this equation gives usconsequentlyand

Here,u( ψ)is the aligned flow parameter.The velocity can be represented as the superposition of an aligned flow and a pure rigid toroidal rotation of each flux surface.Next we substitute equations(2)and(1)into the momentum equation　and　take　its toroidal component to　obtainor

Here,I( ψ)is the intrinsic poloidal current which is not induced by poloidal rotation.Equation(2)shows that the flow is the combination of an aligned flow with a toroidal rotation.The entropy equation guarantees that S=S( ψ).One of the remaining two other components of the momentum　equationisusually　taken as the component along the magnetic field direction,giving(We remind the reader that Γ is the ratio of specific heats.In general,because of the requirement of thermodynamic stability,Γis required to be larger or equal to 1.If heat transport along the magnetic field line is much much(in finitely)faster than the sound wave,then the plasma would have a constant temperature on the flux surface.In this caseΓ=1.In actuality,for a high temperature plasma encountered in fusion plasmas,Γ=1is a very good approximation.For colder plasmas,the transport of heat would not be so much faster than the sound wave,Γwould have a value slightly larger than 1.)This condition can also be written as

Here

H( ψ)is related to the enthalpy of the plasma.The remaining equation is the momentum equation in the direction ofWith the help of the five integral relationships ofu(ψ),Ω(ψ),Hψ),I( ψ)and the entropy function S( ψ),this equation can be written in the form given by Hameiri[16]:

We note that equation(6)is a NLPDE of the poloidal lf ux function ψ.Ifit is elliptic.With source functions on the right-hand side specified inside a given 2D R, Zdomain,ψ can be determined by giving the boundary conditions on the boundary of this 2D domain.Here we are only concerned with this situation.An equilibrium is obtained only after the complete specification of all the five constraint functions{u　,　Ω,I, 　H, S}and the boundary.One interesting point with regard to equation(6)is that the source is proportional to the sum of terms each proportional to the derivative of one of the constraints of{u　,　Ω,I, 　H, S}.The other quantitiesshould all be regarded as functions determined by these constraint functions.The general case is usually solved by using a computer code[18,19].There did not exist a simplified prescription that would facilitate the generation of a systematically variable set of equilibria for the general case.

2.2.Re-examination of Solovev’s approach to the solution of the general equilibrium equation with flow(equation(6))with u=Ω=0.

Solovev’s innovation[1]was to simplify the nonlinear problem given above in equation(6)into a linear problem with linear superposable solutions.Of course,he only dealt with equilibria without flow,i..e.u = 0 and Ω =0.The source constraints were reduced to{I, H, S}.First he simplified the source function so that they were independent of ψ.Also he extended the boundary to the in finite domain.Then he looked for solutions that can be represented in terms of simple powers of R and Z.It resulted in a simple set of functions.It turned out the resultant much simplified set of functions has enough variations to be a useful set.We can attribute his success first to the fact that the governing differential equation(The GS equation for plasma without flow)could be simplified and still satisfy the constraints imposed by{I, H, S}.The second of his success was in parametrizing the source function effectively in terms of a limited number of intrinsic parameters,essentially(maximally)one each for{I　, H, S}.We will call this the source{IP}(intrinsic plasma)set.The simplification process allowed the solution to be composed of the superposition of particular solutions consisting of powers of(R,Z)that were linear with respect to{IP}and homogeneous solutions of the governing equation without source.The amplitudes of the homogeneous solutions we will denote as{OP}(outside plasma).The homogenous solutions gave flux due to the shaping currents(currents from outside of the plasma)and can also be expressed in terms of powers of(R,Z).The number of{IP}was limited by the number of constraints.But the number of{OP}is in finite.An effective means was adopted to arrange them in oder to reflect their importance.By　specifying　all of the　amplitudes　in{P}　　=　{I　P}　　+{O　P}　,we could obtain a solution of the simplified GS equation over the whole space.We denote the total of all these functions with different values of{P}as{Solovev}.Members in{Solovev}consisted of only polynomials inR, Z.A region with a closed boundary specified by　ψ=ψ　bwas then chosen as the plasma region of interest.Only the solution inside this chosen boundary was retained and the outside was discarded.We then had an effective method to construct a library of equilibrium inside a chosen region.Note that this simplification process went beyond linearization.Because he also demanded that the solutions be superposable.Thus the plasma currents are independent of ψ.If plasma current sources are allowed to be linear in ψ,then different source terms could interfere with each other and the solutions are no more superposable.For an actual equilibrium,if the value of ψ was needed for the region outside of the plasma,then a different method,such as the virtual casing principle was used in addition.

As described here,for the solutions to have complete shaping capability,then{Solovev}should at least include all allowed and acceptable solutions of the simplified equation.Physically,we then should be able to describe all possible shapes allowed in the problem.However,in practice,it was found that the actual shaping capability of{Solovev}was limited when we applied them to the study of a plasma with the topology of a torus—showing some possible solutions were not included.Subsequently,in the work of ZWS[2],they found an additional set of source free(vacuum)solutions.The major difference between the ZWS vacuum solutions and those of Solovev was that every member of ZWS vacuum solutions had terms proportional to lnR,specifying source currents on the line of R=0.Although these solutions should not be accepted if our region of interest contained portions of the line,R=0,as the case of the spheromak;they should be included if our final region of interest excluded the R = 0 line,as the case of the tokamak.The difference was that the spheromak and the tokamak belonged to devices with different topologies.Therefore,the concept of completeness in the shaping capability really depended on the topology of the confinement device we were interested in.For devices with the topology of the sphere,the{Solovev}set should have complete shaping capability;whereas for the topology of the torus,we needed to generalize the{Solovev}set to be{Solovev_ZWS}={Solovev} + {ZWS}.Because{Solovev_ZWS}contained{Solovev},of course it also has complete-shaping capability for devices with the topology of the sphere.In the work of Cerfon and Freidberg[7],they noticed this enhanced shaping capability of the{Solovev_ZWS}set and called this property as one-size fits all.

Because of the wide utilization and the enhanced shaping capability of{Solovev_ZWS},and because rotation is such an important aspect of plasma equilibrium,we would like to see whether the approach first pioneered by Solovev can be extended to the case of plasma equilibrium with flow,i.e.for plasma withu≠0,and/orΩ≠0.By extension,we mean that the modified equation and solutions will include the u = 0,and Ω =0or{Solovev_ZWS}as a special case.We will first look at Solovev’s case again here.When we let u = 0,and Ω =0,equation(6)reduces to

And the constraint equation(3)reduces toF=I,and equation(4)reduces toH　(ψ　)　=G　(ψ　. ρ).This implies that ρ = ρ( ψ)andP=P( ψ).A straight forward evaluation of the last two terms in equation(7)reveals they combine to giveWe thus arrive at the usual form of the GS equation with no flow.

同时，电商可以做到个性消费。供应商可以根据消费者的不同要求，一对一地量身订做个性化的产品。消费者可以真正参与到产品的设计、开发、生产等环节，使产品真正做到以消费者为中心，从各个方面满足消费者的个性需求，避免不必要的浪费。

First we notice that the solution depends only on u2and it is independent of the sign of u;meaning that both directions of the aligned flow will have the same solution.Next we notice that there are two positive definite functions ρ( ψ,R)andB2(ψ,R)in addition to the constraint functions.Third we notice the combination ofIt not only enters into the source part of the equation,but also modi fies the basic operator on ψ.This is serious.Because it will affect all the vacuum solutions,i.e.ifwere not a constant independent of both(ψ　,R)all the vacuum solutions have to be modified when there is a poloidal flow.This is an unreasonable situation which we do not wish to happen.Therefore we demand first

The linear-superposition ansatz is accomplished by lettingIn this way the three constraint functions　of{I, H, S}　are　represented　by　the　two{I　P}　　={c　F　,cP}.It is an interesting fact that the two constraints ofH, Scombine together to make P a flux function.The effect of independent variations in the density ρ,which is now also a flux function and entropy S do not enter the equation(8)separately.Of course the full set of parameters are now{cF　　,　cP　　,　ψb　,cv} .Here we used cvto represent all the amplitudes of the vacuum solutions.As mentioned before,by specifying the specific values of the set{cF　　,　cP　,　ψb,　　cv} ,we arrive at an equilibrium inside the domain specified byψb with the plasma current functions specified by　{c　F　,cP}and vacuum source currents specified by{cv}.The corresponding function space we denote as{Solovev_ZWS}.This set should have enhanced shaping capability in specifying the plasma equilibrium for the given plasma current functions.

2.3.Generalization of Solovev’s approach to the case when Ω ≠ 0,u = 0

ForΩ≠0,andu=0,from equation(3)we still have F(ψ)　=I(ψ).And from equation(2)we havevφ=RΩ.From equation(4),we have

晶闸管的额定电压通常选取断态重复峰值电压UDRM和反向重复峰值电压URRM中较小的标值作为该器件的额定电压。晶闸管体的额定电流值通常采用其普通状态时一般电流的2-3倍。在桥式体整流的电压电路中晶闸管内两边承受的最高正反向电压的额定值都是晶闸管正常工作下的额定电压通常采取它的最高正反向平均电压的2-2.5倍。

筛查出30例行为异常儿童。行为异常组儿童和对照组儿童年龄及性别方面差异无统计学意义（P 均＞0．05）。

Equation(6)is reduced to

We note that equation(10)involves the density　ρ( ψ　,R),which depends not only on the poloidal flux function ψ but also on the major radius R.This dependence is constrained by the equation for H given by equation(9).Our question is what is the behavior of the functions{H　,Ω,S}under which the right-hand side of equation(10)can be reduced to a combination of constants and also reduced to equation(8)when Ω=　0.It is quite obvious we need to work on the expression of

We note that equation(9)states that H is a function of ψ only,therefore all the variation with respect to R on the right hand side has to vanish.We can differentiate this equation with respect to R to find out the dependence of ρ with respect to R while holding ψ constant.We thus haveThis equation can be integrated with respect to R to give

From the above subsections,we know that u = 0 and Ω(ψ)≠　0is the only meaningful extension to the Solovev approach.We will concentrate on this case in more detail.Because the strong damping of the poloidal flow arising from the neoclassical effect,this is the most commonly encountered and observed rotation state of the tokamak.For the high temperature plasma,Γ=1is also a very good approximation.Therefore,we will make this assumption here.We note that our discussion here did not start with u = 0 as an assumption,but rather obtained it from a theoretical deduction starting from the basic MHD equations therefore puts it on a much more solid theoretical foundation.With this special type of rotation,we can obtain the GGS equation much more simply.First,the function V = 0 in the expression forand the functionu(ψ)=0in equation(2).Also from equation(2),the rotation velocity can be written as

The function is now the integration constant because we are integrating wrt.to R.Substituting the relationship equation(12)into equation(11)and noting the definition ofP=SρΓ gives us

We now examine the relationship(12).ForΓ　≠ 1,if we letto be a constant independent of ψ.Here R0is an arbitrary radius.Then we have

ForΓ=1,if we letto be a constant independent of ψ,then

We have accomplished our goal of linearization of the equation(6)forΩ≠0.The resultant equation is

with TMdefined through equations(13)–(15).We notice that the introduction of the Mach number M relates the two constraints of Ω andS.These two constraints now cannot have independent values.This is the essence of the simplification process.We conclude that it is totally possible to achieve the goal of simplification if we include a pure toroidal flow into the plasma motion.The result of simplification is to relate the rotation pro file to the density and entropy pro file through the relationship of constant M.Due to neoclassical effect,for confinement system with a long confinement time such as the tokamak,the plasma flow in the poloidal direction is heavily damped.Therefore,this is an important situation.In particular,the plasma behaves more like a system withΓ=1.We will discuss the solutions in more detail in a later section.

2.4.Generalization of Solovev’s approach to the case when u ≠0;Ω =0 or Ω ≠ 0

We first start with the case ofΩ=0.Ifu　≠　0,　Ω=　0,the plasma has only an aligned flow,without an additional toroidal flow.In this caseThen equation(10)can be written as

小学数学教师在使用多媒体技术进行教学辅助过程之中，必须合理地安排多媒体技术与传统教学法之间的关系。也就是说，小学数学教师首先应当看到多媒体技术对于数学课堂教学的优势所在，同时，教师还必须意识到的是：将多媒体技术融入到教学设计的目的就是为了更好地达成教学目标。在此，教师特别要关注学生的发展，设计新颖的模式。多媒体教学虽然具有了上述种种表现手法，但若使用不当，极易使学生形成“等着看”的惰性心理，因为我们有很多的教学课件，总是以“问题—探索—结论”的模式出现的。所以，作为教育的主导者、设计者，我们的课件必须是有新意的、必须是出乎学生意料但又合乎情理的、必须是有利于教学目标达成的。

Equation(18)is quite a stringent physical condition.Because it demands that asu　i.e.we start with a plasma without either pressure and density when the poloidal rotation vanishes.Nevertheless,we can solve for B2in terms of all the other constraints by using the enthalpy relationship giving

With the requirements of equations(18)and(19),the constraint H becomesThe equilibrium equation can be renormalized by dividing out the factor ofto become

In equation(20),we integrated out the integral and noting that the integration constant has to be 0,since ρ is proportional tou2(ψ).We recognize easily in this case,the recipe as given previously for the process of generalization of the Solovev linearization involves requiringand for the last term.These relationships should hold even for u　2=　0.Therefore,cP　=　0.We arrive again at the condition of a pressureless plasma.Unless we are interested in modeling the poloidal rotation of a pressure-less plasma,this would not serve the purpose of generalization to the Solovev process.Therefore,this is not an interesting case.

（3）通过对甘肃某金矿开展选型试验知，利用产品细磨变化曲线和能耗关系，可以选定装机功率280kW、容积10.0m3的KLM-280立磨机；实际应用表明，设备可完全满足矿山选厂的要求。

Section　4　is　on　complete-shaping　capability　of{SOLOVEV_ZWSm}.We first showed that the completeshaping capability of the set is related to the topology of the plasma con figuration.For plasmas with the topology of the sphere,should have complete-shaping capability.are not admissible solutions because it would　mean　singular　currents　in　the　plasma.is incomplete in its shaping-capability without including the setif we are interested in plasma with the topology of the torus.Becausebecome allowed solutions for plasma topology of the torus in which the center φ axis is not part of the plasma.We next showed that{SOLOVEV_ZWSm}still does not have complete-shaping-ability for plasma with boundary having negative curvature points over part of its boundary.This is the case of the doublet and the bean shaped plasma.

Because the case ofΩ≠0should includeΩ=0as a special case.We also conclude that the Solovev simplification process can not be extended to situations involving a poloidal rotation,except possibly the uninteresting case of a pressure less plasma.

总之，印尼语的地位逐渐提升，成为地位最高的语言，使印尼逐渐成为多元文化的“单语国家”。华语地位从荷印殖民时期的华人民族共同语变成新秩序时期的非法语言，到目前成为具有外语法律地位的印尼华族的共同语。

2.5.The maximal linear problem for the case of Γ=1,Ω（ψ）≠ 0,u = 0

紫杉醇涂层支架应用于临床中，对中期近期均能够起较好的应用效果，对再狭窄情况加以预防，但临床中针对其远端效果研究还并未抑一致。现今临床的改良紫杉醇涂层支架工艺，对支架的材料、形状有效调整，在减小支架刚性的同时增强了弹性，发现可以有效抑制内膜增生减少损伤。笔者也相信在当前临床医疗技术不断创新背景下，冠心病介入治疗应用紫杉醇涂层支架必将具有更广阔的应用前景。

where Ω is the toroidal rotation angular frequency of the plasma,which is a function of the flux surface.We may regard　ρ= ρ( ψ　,R)andP　=　P (ψ　,R),then we decompose the momentum conservation equation into components inanddirections.The force balance in the direction ofcan be written as

BecauseP　=　S　(ψ) 　ρ　=　Tρ,the plasma temperature is constanton　the flux　surface　and　we　can　integrate equation(22)to obtain

Hereis the square of the Mach number,T is the temperature and m the mass of ions on the flux surface and Rais the integration constant and is an arbitrary function for each flux surface.We adopt a different(from that of equation(12))form, ρ¯,because this will help us utilize the simplification process and facilitate the description of our solution space.Also,because of the fact that the integration constants can be chosen differently so long as we understand its implication.Here,are the values of[ρ,P]atR=Ra.We assume our region of interest contains a magnetic axis.Ra is chosen to be the value of R of the magnetic axis.The force balance relation in the direction ofthen gives

Equation(24)agrees with that given first by Mashke and Perrin[20].At this point,the functions on the right-hand side,are the assumed(nonlinear functions)given source functions.Racan be regarded as given or chosen.In general,given a closed 2D domain,this is a nonlinear elliptic partial differential equation for the poloidal flux function ψ.This 2D domain is first defined with boundary condition ψ=ψ　bspecified on the boundary.We are therefore looking for the solution of a nonlinear 2D boundary value problem defined by the partial differential equation(24).It can be solved quite readily by using a computer[18,19].We are still at a completely general situation.

Now we start the Solovev simplification process.First,we assumeFF′,are constants.The NLPDE is reduced to a LPDE with constant source terms with superposable solutions.Second,because of the linear nature of the problem,we can obtain first all the solutions of the linearized equation(24)over the whole space.Now thatthere are only two source terms on the rhs of equation(24).The solution space consists of functions containing two special solutions each due to one source term and an in finite set of solutions with no source(in the finite domain).The implied sources are outside of our domain of interest.These outside sources to be chosen depending on the coefficients of the vacuum solution.We denote this solution space as{Solutions}.This variability of choosing solutions from{Solutions}is the major advantage of the Solovev approach.It is arrived at by sacrificing all the details of the nonlinear FF′,functions and retaining only their constant approximations.In Solvev’s original treatment,he allowed only polynomial functions.Therefore it was actually incomplete.We should include all functions that are physically allowed(that does not have flux and current singularities)in our yet unspecified plasma region of interest.This really covers the whole poloidal plane minus the boundary.Therefore,we lost the boundary at a finite value in the boundary value problem.The boundary has been pushed out to the boundary of the poloidal plane.This allows us to impose extra conditions to fix our set of functions to suit our purpose.The next step is to choose all the coefficients of the vacuum solutions.This means taking out a particular memberψsol(the subscript sol here stands for solution)from{Solutions}.Now that we have a definite value of ψ defined by ψsol.The next is to fix our plasma region of interest by choosing a value ofψb.The flexibility of choosing the amplitudes of the vacuum solutions is the main reason that this process can accommodate quickly various shapes.

We first rewrite the linearized equation(24)again

The solution space can be more cleanly defined by first taking out the physically scalable parameters.We first normalize the length scale wrt.Raand letandWe also normalize the source density with respect to its value at Raand denoteandNote that this parameter βpJis a force-balance based definition ofβpand is different from the usual definition of energy-content basedβp.This fixes the normalization for ψ as .In summary

Using the above dimensionless parameters,we rewrite equations(24)as

And the equilibrium equation is reduced to the following simple form:

Equation(28)has therefore only two essential parameters,βpJandc　(=　M2).These are the internal parameters,or{IP}={β　　pJ,c}.This should be contrasted with the essentially in finite number of parameters if we we2re to retain the full nonlinear variations in theFF′,and Mfunctions.The{OP},which specifies the amplitudes of the external current source actually remains the same as the case with no flow.The modification due to toroidal flow with constant Mach number introduces just one more dimension to the solution space as expected.With these normalizations,the magnetic axis is now at(x　, z)　=(1　,0).We further normalize the poloidal flux function to be 0 at the magnetic axis.Then the internal boundary condition for the equilibrium equation is

We note that the problem is now completely linear with superposable independent solutions.We emphasize here that not only equation(28)is linear,the internal boundary conditions equation(29)are also linear.An efficient categorization of the solution should fully utilize the linearity of both equations(28)and also(29).From one particular solution,with the particular source function,and satisfying the internal boundary conditions equation(29),we may generate others by adding any solutions of the homogeneous problem which will also satisfy the internal boundary conditions without having to rearrange the coefficients to satisfy the internal boundary conditions.This could have been but was not done in the original solution provided by ZWS[2].We do not have to specify the shape of the plasma at this point.Its determination is decided by our choice of the homogeneous(vacuum)solution.Therefore,the most general set of vacuum solutions should give us the most general shaping capability or the most general shapes of the plasma.We arrived at obtaining the{SOLOVEV_ZWSm}space.

3.General solutions,plasma shapes and the equilibrium parameter space

In this section,we present the independent solutions of equation(28)together with the implementation of the internal boundary conditions equation(29).Of course,the totality of our solutions is the same as the set of functions given by Solovev and ZWS.Each member of our set is just a linear combination of the{Solovev-ZWS}set,but the members of the two sets are different.There is an expected difference in the easiness in the actual application of the members of the two sets.In some situations,the set given by Solovev and ZWS could be easier and more appropriate.But for the problem at hand as we defined it here,we feel that to use the set presented by us could be much simpler.The behavior of the different members at special location are then studied next.Depending on the particular nature of the system under consideration,some of the members are then excluded.This results in that the complete-shaping capability of the system would actually depend on the actual system under study.This dependence is discussed next.

综上所述，临床护理甲亢合并2型糖尿病患者时，通过护理干预的开展，可有效改善患者甲状腺功能，降低血糖水平，预防疾病进展，减少相应并发症，并改善预后，提高患者的生活质量。

3.1.The general solution of equations(28)and(29)

The purpose of our present work is to re-examine the approach in generating{Solovev-ZWS}and make extension to toroidally symmetric plasma with a general flow.Starting from the full MHD equations,our study first showed that the only meaningful extension possible is to plasmas with a pure toroidal rotation and with a constant Mach number.This results in a relatively minor extension.Secondly,we point out that there should be a slightly better formulation of the problem to fully take advantage of the simplified solutions.In the Solovev-ZWS approach,the plasma is first assumed to cover the whole space,therefore,at this stage,there is no boundary nor any boundary conditions.The GS equation actually allows us to set an internal boundary condition,i.e. fix the value of the solution at a specific point.For the closed configuration at hand,we propose to fix the value of the flux function at the(elliptic or hyperbolic)magnetic axis as our internal boundary condition.With this added requirement,our set of functions represent a recombination of the functions in the{Solovev-ZWS)set.We will call this the{Solovev-ZWSm}set.Third,there are situations,for instance the doublet[22]or the indented bean-shaped tokamak[23,24],that require an intrinsic current source near the plasma boundary to maintain the con figuration.This intrinsic location could be either near or far from the central coordinate axis where the singular currents in the{Solovev_ZWSm}set is located.This situation complicates and reduces the additional shaping capability provided by{ZWS}for the doublets or indented tokamaks using{Solovev-ZWSm}.We need to keep this in mind when we utilize this set of functions and the shaping capability of the{Solovev_ZWSm}remains incomplete for these con figurations.

时令美馔不胜数。通过《东京梦华录》《都城纪胜》《西湖老人繁胜录》《梦粱录》《武林旧事》等两宋典籍得知，宋朝人口福多多：面食有环饼、油饼、白肉胡饼、莲花肉饼、炊羊胡饼、天花饼、烙饼、馒头、素饼（面条）、焦碱水锥（炸元宵）、浮团子（汤圆）、角子（饺子）；肉食有连骨熟肉、爆肉、肉脯和干肉；羹汤有缕肉羹、肚羹、玉糁羹和各种各样的“汁水”等。繁华的宋朝，真是让老饕羡慕嫉妒恨的时代！

with

We demand that each of the components　ψF　,　ψp　,ψvac satisfy the conditions imposed by the boundary conditions equation(29),then the solutions are uniquely defined.

3.2.The special solutions

It is easy to verify that the following are the special solutions:

We notice that if we letc→0in equation(32),then we have Of course,the special solution is really not unique.The choice we made in equations(31)and(32)is such that when c = 0,and without the addition of other vacuum solutions,the solution will approach a shape with a circular cross-section near the magnetic axis.These choices incidentally also gave natural shapes to the plasma at larger minor radius even without any additional external fields.This behavior of solutions with natural shapes depending on the aspect ratio of the plasma is a direct consequence of the toroidal symmetry.These shapes also have a definite dependence on the Mach numbers when there is a finite toroidal flow.We also note that,as is well known,the expression given forψFcan not be used for systems with the topology of the sphere.In other words,for systems such as the spheromak,we have to let　β　pJ=1.

3.3.The vacuum solutions

We show in appendix A and in more detail that the most general homogeneous solution is given by

Here,eachandis up–down symmetric solution if N is even,and up–down anti-symmetric if N is odd.Each one of them satisfies the source free GS equation equation　(28)　and　the　internal　boundary　condition equation(29).is defined in equation(44)in terms of the primary un-normalized function fN,which is defined in turn by the polynomial expression given in equation(39).The coefficients in the power series in fNsatisfy the recursive relationship given in equation(40).tends to large values when.But they remain finite and without current singularity when(x,z)are small.Therefore,they represent the poloidal flux produced by shaping currents outside the domain of interest and located far away from the coordinate center.is defined in equation(45)in terms of the primary un-normalized function fNmultiplied with lnx,and another unnormalized function gN,which is defined in turn by the polynomial expression given in equation(41).The coefficients in the power series in gNsatisfy the recursive relationship given in equation(43). tends to large values also when,it also contains current singularities at the center φ line.Therefore,they represent the poloidal flux produced by shaping currents outside the domain of interest and located both far away but also very near the coordinate center.These are therefore related to the internal shaping fields that could be used specifically to shape the inboard side of the plasma shape.aN,inandaN,outare their corresponding amplitudes.Some of the first few of andare:

将Fs=325 kN,A=πd2/4,[τ]=164 MPa(销轴材料为42CrMo,其抗拉强度为1 080 MPa,考虑到轴承工作重要性、疲劳、计算误差、制造可靠性等问题,确定安全系数取3.96,确定许用强度为273 MPa,进而确定许用剪切强度为0.6×273=164 MPa)代入式(4),得d≥58 mm.根据轴承手册选用轴承GEF60ES,其额定动载荷为55 kN,额定静载荷为1 720 kN,满足要求.

3.4.Plasma shapes

Figure 1.The basic parameters of a typical plasma equilibrium con figuration that determine the shape of its cross-section in the poloidal plane.The major radius has been normalized to 1.

A given equilibrium is determined by specifying all the coefficients{aN,in}and　{a　N　,out}.　Of course,for relatively smooth shapes,most of the higher ones are usually chosen to be 0.The plasma shape is then determined by choosing a value for the plasma boundaryψb..In order to investigate the shapes of this last magnetic surface(all flux surfaces inside of this boundary are also determined and have different shapes),we de fine the shape factors strictly based on the last surface as follows:(see figure 1)

Figure 2.Various elongation of the plasma achieved for a plasma with a　=　βpJ=　c　=　0.5by fixinga2　,in=　0.25and varying the coefficient a4,out.

Figure 3.The triangularity of−1 achieved by using a2,inonly and triangularity of 1 achieved by using a4,outonly for a plasma witha　=　βpJ=　c　=　0.5.

where x　max　,zm　ax　,x minand zminare the maximum and minimum of x and z of the closed boundary magnetic surface,separately.xmax,inboardis the maximum value of x for the inboard side of the boundary.A is the cross-sectional area of the magnetic surface,andA　e=πa　b,A　t=　2abandAr=4　ab.κ,δ(u,d),δsand i are the elongation,(upper,lower)triangularities,squareness and indentation of the equilibrium.We show in figure 2.the elongation achievable for a plasma witha　=　βpJ =　c　=　0.5by fixing the value ofa　2,in=0.25 and varying the coefficients ofa4,out.

We show in figure 3 the triangularity achievable for a plasma witha　=　βpJ=　c　=　0.5.If we use the coefficient of a2,inalone,we can reach value of triangularity of−1;whereas if we use the coefficient ofa4,outonly we can reach triangularity of 1.

It is seen that using{Solovev_ZWSm}we can achieve rather extreme shapes showing the extensive shaping capability of this set of functions.

3.5.The parameter space

There is ordered correlation of influence of values of{ψ　　　b　,　β　pJ,　aN　　　,in,　aN,out }on the values of　{a,κ , δ, δs,…}.In general,the two basic quantities{a　,βp}are related mostly to{ψ　　　b　,βpJ}.However,because each quantity is also related through their definitions to the solution chosen,the relationship is not a complete one-one correspondence.Other parameters will have their influences,albeit less so on them.Intuitively,we have a semi-hierarchy of importance attached to the shaping parameters,i.e.{　　κ　 , δ, δs, …}Then we can find the parameters in{aN　,in,aN,out}that have the most influence on　κ , δ, 　δs, etc.This gives us a semi-hierarchy and allows us to search through the shape parameters efficiently.This point has been studied extensively through numerical computations and will be reported separately.

4.Relevance of the{Solovev-ZWSm}functions to different confinement systems

4.1.Behavior of the solutions at special locations and admissible solutions for different confinement topology

With the inclusion of finite toroidal rotation,c≠0,although locations atρ→∞are never intended to be within the plasma,the solution given in equation(32)indicates that the behavior of ψ would be dominated by the special solution due to rotation.If we concentrate on the vacuum part of the solution,we can first solve the vacuum equation using spherical polar coordinates.This is done in appendix B.The result is given in equation(52).For the choice of the functions we made for the set{Solovev_ZWSm},the radial dependence ofshould be like　ρN,and the angular dependence of the solution should behave like Near the center linex=0,　is well behaved,including all its derivatives.But for,it contains a termx2 　zN　-2　lnx,we notice that althoughandare all still well behaved,the magnetic fielddevelops alnxsingularity.Hence the current develops asingularity.Thus these　should be excluded if the eventual plasma region contains the x = 0 line.The confinement systems with the topology of the sphere are spheromaks or pinches.For these con figurations,the set{Solovevm}is sufficient for us to explore its equilibrium space.

The confinement systems with the topology of the torus include the tokamaks and its many variations,RFP’s,the hard core pinch,and so on.In the present context,the Bumpy Z-Pinch[25],a ball shaped pinch with electrodes placed at the top and bottom of the central region for helicity injection,should also be classified as having the topology of the torus.For these con figurations,the full set{Solovev_ZWSm}is needed for us to explore its equilibrium space.We note the important contribution by ZWS[2]in giving the complete vacuum solutions.We also wish to point out that members of{Solovev_ZWSm}are combinations of the solutions found by them.Efficiency is gained by incorporating the conditions in equation(29)and use the exact members in the set of{Solovev_ZWSm}.

4.2.Lack of complete-shaping-capability of{SOLOVEV_ZWSm}resulting from essential vacuum singularity of the configuration

In this subsection,we wish to show that depending on the property of the equilibrium we are looking for,even for con figurations with the topology of the torus,the set{Solovev_ZWSm}could potentially be incomplete.Consider the case of the doublet[22]or the indented bean shaped tokamak[23,24].These shapes have negative curvature on its boundary and necessarily require current(s)at a finite location near its boundary to maintain it.The locations are approximately less than from the plasma boundary.

Therefore,we conclude that for con figurations with κp(s)　　＜　0on　portion　ofits　boundary,the　{SOLOVEV_ZWSm}set is not expected to have complete shaping capability.Hereκpis the curvature of the point on the plasma boundary.An explicit example of this required external essential current was given in the work of[26].{Solovev_ZWSm}has all the singular shaping currents placed at the boundary of the in finite plane or the central symmetry axis.Aside from the usual function of producing to the usual desired shaping,such as κ, δ,the singular currents now have to serve the additional function of providing the indentation to the plasma boundary.There is not enough flexibility for the coefficients of the vacuum solutions to ful fill this extra function.The shaping capability is therefore lost.As an example,we show in figure 4 the maximum elongation achievable for a doublet with indentation parameter i=0.5 by using{Solovev_ZWSm}.It is seen that for low aspect ratio doublet,we can only achieveκ=5;whereas for a large aspect ratio doublet,we can achieve a maximum elongation beyondκ=15.

本文以热压R1、R6、R10、无压W1胎体、复合片为焊接基材，以45钢为焊接母材，选择5种不同牌号的银钎焊料，对其焊接强度试验进行设计研究。银钎焊料主要牌号如表1所示。

5.Summary

In this work we studied the possible generalization of Solovev’s approach[1]to the toroidal equilibrium problem to plasmas with arbitrary flows.

We showed in section 2 that although Solovev’s approach started with simplification of the source pro files,it also fundamentally modified the solution procedure in obtaining the allowable solutions.This led to its great Efficiency.We started with the general equilibrium equations for a tokamak with arbitrary flow.Following the derivation initially given by Hamieri[16],the general equilibrium problem is reduced mathematically to a boundary value problem of solving the NLPDE(6)for the poloidal flux function ψ with the five source functions related to the aligned flow parameter u( ψ),given in equation(2);the toroidal angular rotation frequency Ω(ψ);the intrinsic poloidal currentI( ψ),given in equation　(3);the　specific　enthalpy　H(ψ),given　in equation(4);and the entropy functionS(ψ).We next check the compatibility of these constraints with the simplification assumption of the Solovev’s approach.We showed that Ω　(ψ　)　=　u　(ψ)　=0corresponds to Solovev’s original case.We also found that foru　(ψ　)　=0,　Ω(ψ)　≠0,simplification assumption demands that the toroidal Mach number be a constant across the plasma.There is no restriction on the value of the specific heat capacity Γof the plasma.We next found thatu(ψ)≠ 0is not compatible with the requirement of a desirable generalization.The main reason is thatu≠0 affects the partial differentiation operator.Except for the uninteresting case of a zero density plasma,the generalization to theu≠0case is not possible.

Figure 4.Left:the doublet plasma with i　=　0.5,a　=　0.8.Right:the maximum achievable elongation with different inverse aspect ratio a.At low aspect ratio,onlyκ=5can be achieved,whereas at small a,κ can reach beyond 15.

We next examined the solution procedure.In the ordinary approach,a specific equilibrium,starting with a given boundary∂D,the equilibrium problem requires first choosing ψ　bon ∂D,then find ψ(R, Z)inside∂D　by solving equation(6).Therefore,it is to find solution for a NLPDE inside of a finite domain.Solovev’s approach relied on changing the basic nature of the problem.By making the assumption of constant current density pro file,Solovev turned the nonlinear equation equation(6)into a linear problem with source functions that do not depend on ψ.This allowed the solutions to be first obtained over the entire in finite plane and also allowed the solutions to be composed of its independent superposable parts:a few particular solutions,each depending on a specific source;and an in finite set of source free(vacuum)solutions.The choice of anyψbthen gives a new solution for plasma with the same source pro file but with the boundary given by　ψ=ψ　b.Changing the domain of the solutions allowed us to impose internal boundary condition.This could be utilized to our advantage as shown in equation(29).

The general solution of the equilibrium problem for a toroidal plasma with constant pressure gradient,current and Mach number pro files are given in section 3 for the most important case ofΓ=1.The solution consists of the superposition of special solutions due to plasma current sources{ψ　F　,ψ P} and external vacuum solutions with poloidal flux due to current sources far away from the plasma(the shaping fields).The special solutions are chosen to represent a large aspect ratio circle near the elliptic magnetic axis for the plasma without taking into account of external shaping.The vacuum solutions consist of two sets of functions.The first set represents the current sources far away from the plasma.This corresponds to the original vacuum set used by Solovev(with a trivial generalization).The other setrepresents currents not only from the far away boundary but also on the central φ axis.was first obtained by ZWS[2].The combined set of functions{SOLOVEV_ZWSm}=represents the most general possible generalization to the Solovev type of equilibrium for a rotating plasma.Although the total function space of{SOLOVEV_ZWSm}is the same as that of Solovev and supplemented by the ZWS solutions,each member in{SOLOVEV_ZWSm}is slightly different from that originally used by Solovev and ZWS.The reason is because we now included the internal boundary condition for each member solution.

三是，内容编排顺序存在差异．具体而言，苏教版编排的4个例题是“分数除以整数”“整数除以几分之一”“整数除以几分之几”及“分数除以分数”，其具体展开方式如下．

The steps taken in obtaining the setsandare given in appendix A.They are defined as power series in(x, z).The relationship ofto the usual solution of the Poisson equation in spherical coordinatesis given in appendix B.This allowed us to have a closed-form representation of its angular dependence for members in

This work was supported by the program of Fusion Reactor Physics and Digital Tokamak with the CAS ‘One-Three-Five’Strategic Planning,National Natural Science Foundation of China under Grant Nos.11375234,11105175 and 11475219 and also partly by National Magnetic confinement Fusion Science Program of China under Contract Nos.2015GB101003 and 2015GB110001.The authors would also like to acknowledge the ShenMa High Performance Computing Cluster at the Institute of Plasma Physics,Chinese Academy of Sciences.Part of the present work is revised from a previous unpublished manuscript sent to another journal.We would like to thank the referee of this other journal for pointing out to us the works of Zheng-Wootton-Solano[2]and Cerfon and Freiberg[7].This led us to improve our presentation.

Appendix A.Derivation of the ψoutand ψinfunctions

The vacuum flux function satisfies

Becauseψvacrepresents poloidal flux due to sources at the coordinate boundary away from the plasma region of our interest centered around(x　, z)　=　(1　,0),the solutions should in general be increasing away from(x　, z)　　　=(1　,0).We first seek polynomial solutions fNin uniform power N of x and z for the above equation.Therefore

Substitution of equation(39)into(38)gives the recursive relation

In order for the sequence of coefficients to terminate,we have to choose forN=even,a0=1,a1=0;and forN=odd,a0 = 0,a1=1.We thus obtain the following expressions for fN:

We notice that the sethas no current singularities at x=0.We will identify the set includingas the modified set{SOLOVEVm},a generalization of the set given by Solovev to plasma with toroidal rotation.Because if we letc →0,except some relatively minor modifications to the vacuum solutions with no change to their singularity behavior,we arrive at the solutions proposed originally by Solovev.The subscript m stands for incorporating the boundary conditions equation(29).{SOLOVEVm}actually has complete-shaping capability for confinement systems with the topology of the sphere.For confinement systems with the topology of the torus,now that the line x = 0 is outside of the plasma domain,we need to use the set{SOLOEV_ZWSm}=　However{Solovev_ZWSm}still does not have complete-shaping capability for plasmas with the topology of the torus.We comment here that the concept of complete-shaping capability of the set of functions really depends on the topology of the system and the curvature vectors of the plasma boundary(see section 4.2).

These solutions tend to large values when(x,z)become large.They have the correct behavior there.However,none of the solutions has singularities in its derivatives whenx→0.They cannot represent fields produced by currents on the φ axis.We would expect that these vacuum solutions are not complete.We expect another set of solutions which could develop a current singularity when x becomes small,if we are interested in confinement systems with the topology of the torus.Because the x small region would then be outside of the plasma.Note that because the recursive relation equation(40)connectsam+2with amand skippeda　m+1,there is indeed another set of solutions which behaves aswhere gNis a polynomial of degree N in x and z.Substitution of this expression into equation(38)gives us

We first note that f0and f1are special.The 0th order and first order polynomials spaces do not allow other solutions.Therefore,the additional solution starts with N = 2.All the fNends with a term proportional to in equation(41)does not have a negative exponent.The number of terms with nonzero coefficients are for N = even and for N = odd.We also need only a special solution for gN.Therefore,we may choose our representation for them as

wherem　0=　2if N = even andm　0=　3if N = odd.Substitution of this expression into equation(41)then gives us the upward recursive relation,(starting withb　0=　0,orb1　=　0in the following equation):

By using the above recursive formula equation(43),we obtain the following expressions for the

It is interesting to notice that because each of the gN function contains the factor z2,they all satisfies the internal boundary condition equation(29).But the two independent solutions fNand hNdo not necessarily satisfy the imposed boundary conditions at the magnetic axis because of fN.This arises from the fact the magnetic axis is chosen to be at(1,0).Or for the magnetic axis to remain fixed,proper external radial and vertical fields need to be imposed for the new vacuum solutions found.ThefN,hNvacuum solutions can be easily ‘renormalized’to become vacuum solutions that satisfy the internal boundary conditions at the magnetic axis.Therefore we need to obtain

It is easy to inspect the resultant expressions and find them using the recursive relations equations(40)and(43).Therefore,these vacuum solutions start with N=2.For N even,the expressions are up–down symmetric in z;and for N odd,they are up–down anti-symmetric in z.Becausegoes to large values only when(x,z)tends to large values and has no current singularity when x becomes small,we will identify as the outer vacuum solutions Whereascan develop current singularity when x becomes small,wewill designateas the inner vacuum solutions.After renormalizationand.Some of the resultant functions are then listed in section 3.

Appendix B.Solution of the vacuum equation in polar coordinates

Consider solving the vacuum equation in equation(30)in the{ρ , θ}coordinates wherex　=　ρ sinθ,z　=　ρcosθ.It is then

This is a situation where we can use the method of separation of variables and ifwe will have

We can compare this equation with the usual toroidally symmetric Poisson　equation　which　can　be expressed as

We can similarly use the separation of variable ansatz,with the well known situation of,forwe have

In equation(49),l≥0,is any integer.This gives the general solution for Φ as

In equation(50),are arbitrary constants.The are the Legendre polynomials of the first kind;and thes are Legendre functions of the second kind.EachQl( c　osθ)contains an expression of the formand it thus contains logarithmic singularities atWe notice that if we operate on equation(48)withwe obtain

Comparing equations(51)and(46),we recognize from the angular variation part of the equation that the constants of separation should be the same and can immediately write down the solution ofψvacas

In equation(51),are arbitrary constants.Now,we demand that atis not more singular than the logarithmic function.ThenIt is interesting to note that although the portions of the expressions containing Ql contain the factorit is never singular.But its behavior when multiplied into　l ρwill not behave like a power series in(R,Z).Now if we look for solutions that behave like power series in(R,Z),we can identify

References

[1]Solov’ev L S 1967 Zh.Eksp.Teor.Fis.53 626 Solov’ev L S 1968 Sov.Phys.—JETP 26 400

[2]Zheng S B et al 1996 Phys.Plasmas 3 1176

[3]Wan B N,Ding S Y,Qian J P,Li G Q,Xiao B J and Xu G S 2014 IEEE Trans.Plasma Sci.42 495

[4]Chan V S et al 2015 Nucl.Fusion 55 023017

[5]Wandong Liu et al 2014 Plasma Phys.Control.Fusion 56 094009

[6]Shafranov V D and Zhakharov L E 1972 Nucl.Fusion 12 599

[7]Cerfon A J and Freidberg J 2010 Phys.Plasmas 17 032502

[8]Brau K et al 1983 Nucl.Fusion 23 1643

[9]Graves J P,Hastie R J and Hopcraft K I 2000 Plasma Phys.Control.Fusion 42 1049

[10]Kissick M W et al 2001 Phys.Plasmas 8 174

[11]Betti R 1998 Phys.Plasmas 5 3615

[12]Bhattacharyya S N 1994 Phys.Plasmas 1 614

[13]Miller R L et al 1995 Phys.Plasmas 2 3676

[14]Bondeson A and Ward D J 1994 Phys.Rev.Lett.72 2709

[15]Chu M S et al 1995 Phys.Plasmas 2 2236

[16]Hameiri E 1983 Phys.Fluids 26 230

[17]Tasso H and Throumoulopoulos G N 1998 Phys.Plasmas 5 2378

[18]Guazzotto L,Betti R,Manickam J and Kaye S 2004 Phys.Plasmas 11 604

[19]Yanqiang H,Yemin H and Nong X 2016 Phys.Plasmas 23 042506

[20]Maschke E K and Perrin H 1980 Plasma Phys.22 579

[21]Poulipoulis G,Throumoulopoulos G N and Tasso H 2005 Phys.Plasmas 12 042112

[22]Ohkawa T and Jensen T H 1970 Plasma Phys.12 789

[23]Miller R L and Moore R W 1979 Phys.Rev.Lett.43 765

[24]Chance M S,Jardin S C and Stix T H 1983 Phys.Rev.Lett.51 1963

[25]Jensen T H and Chu M S 1981 J.Plasma Phys.25 459

[26]Chu M S and Dobrott D 1973 Phys.Fluids 16 294