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The EPSILON experimental pseudo-symmetric trap

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2001 Nucl. Fusion 41 945

(http://iopscience.iop.org/0029-5515/41/7/315)

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The EPSILON experimental pseudo-symmetric trap

V.V. Arsenin, E.D. Dlougach, V.M. Kulygin, A.Yu. Kuyanov,A.A. Skovoroda, A.V. Timofeev, V.A. Zhil’tsov, A.V. ZvonkovNuclear Fusion Institute, RRC Kurchatov Institute,Moscow, Russian Federation

Abstract. Within the framework of the Adaptive Plasma Experiment (APEX) conceptual project, a

trap with closed magnetic field lines, the Experimental Pseudo-Symmetric Closed Trap (EPSILON),

is examined. The APEX project is aimed at theoretical and experimental development of the physical

foundations for a steady state thermonuclear reactor designed on the basis of an alternative mag-

netic trap with tokamak-like large β plasma confinement. A discussion is given of the fundamental

principle of pseudo-symmetry, which a magnetic configuration with tokamak-like plasma confinement

should satisfy. Examples are given of calculations in the paraxial approximation of pseudo-symmetric

curvilinear elements with a poloidal modulus B isoline. The EPSILON trap, consisting of two direct

axisymmetric mirrors linked by two curvilinear pseudo-symmetric elements, is considered. To increase

the equilibrium β, the plasma currents are short-circuited within curvilinear equilibrium elements. An

untraditional scheme of MHD stabilization for a trap with closed field lines by use of axisymmetric

mirrors with a divertor is analysed. The experimental installation EPSILON-One Mirror Element

(OME), which is under construction for experimental investigation of stabilization by divertor, is dis-

cussed. The opportunity for applying the ECR method of plasma production in EPSILON-OME in

conditions of high density and low magnetic field is examined.

1. Introduction

The concentration of intellectual efforts and finan-cial resources on tokamaks has made possible theexperimental proof of the practicability of controlledfusion in systems with magnetic confinement of hotplasma. The economic and technological realizationof a thermonuclear reactor has begun.

A basic question in the physics of magnetic con-finement has acquired a new urgency. Is it possible, inprinciple, to significantly improve the properties of afusion reactor? The improvements to be achieved arewell known: large β, steady state operation, and useof D3He, a low activity fuel. The Kurchatov InstituteAdaptive Plasma Experiment (APEX) conceptualproject is aimed at the development of the physicalfoundations for a steady state fusion reactor basedon an alternative type of magnetic trap that wouldmake it possible to achieve high β with a tokamaklevel of plasma confinement time. In APEX, vari-ous steady state magnetic configurations with largeβ and without superbanana charged particle trajec-tories are examined. In the present article the resultsof development of the idea of linked open traps arepresented.

It is well known that plasmas with large β canbe obtained in mirrors, but with short confinementtime compared with that of tokamaks. Ambipolaropen traps demonstrate good hot ion confinement,

but the electron temperature is low. An obvious solu-tion to the problem consists of linking two mirrors bytwo curvilinear elements. B.B. Kadomtsev discussedsuch a possibility at the beginning of thermonuclearresearch [1]. Many authors offered variants of linkedmirrors. The ELMO Bumpy Torus experiments arethe best known (see the last proposal in Ref. [2]). InRef. [3] a linked mirror neutron source was proposed.The very interesting trap DRACON, with rotationaltransform, was proposed in Ref. [4]. The authors dis-covered a curvilinear equilibrium element with short-circuited plasma currents. At present there are noexperimental linked mirrors. The physical reason forthis is the large plasma losses in all proposed mag-netic systems [1–4]. One of the basic APEX tasksis to explain the fundamental reasons for the largelosses and to develop proposals for their elimina-tion by modifying the geometry of the magneticconfiguration.

2. Principle of pseudo-symmetry

The search for the optimum magnetic configura-tion from the standpoint of steady state confinementof a hot, high pressure fusion plasma is mainly basedon the analysis of the topography of the magneticfield strength B on an equilibrium magnetic sur-face [5–9]. There are no superbanana losses if allB = const contours on the equilibrium magnetic

Nuclear Fusion, Vol. 41, No. 7 c©2001, IAEA, Vienna 945

V.V. Arsenin et al.

surface encircle either the magnetic axis or the majoraxis of the torus. The first case is called poloidalpseudo-symmetry (PP), the second toroidal pseudo-symmetry (TP). The simplest example of a PP typetrap is axisymmetric mirrors. The tokamak is thesimplest TP type system. The two types of pseudo-symmetry are topologically incompatible. The prin-ciple of pseudo-symmetry is that when joining theseparate traps only one type of pseudo-symmetryshould be used. All proposed linked mirrors contra-vene this principle, and as a result the confinementis poor.

In the most general form the condition for pseudo-symmetry is as follows [7]:

(B ×∇ρ) ·∇BB ·∇B = f (1)

where f is a bounded function, B is the magneticfield vector and ρ is an arbitrary magnetic surfacelabel. The magnetic surfaces in a trap without rota-tional transform are defined by the well known rela-tion (see, for instance, Ref. [10])

U = −∮dl

B= const. (2)

The choice of function f affects the plasma con-finement through a change of magnetic field geom-etry. For instance, the choice f = 0, known asthe isodynamic or orthogonality condition, makesthe configuration an ideal geometry for plasma con-finement, because the drift surfaces of all chargedparticles coincide with the magnetic surface. Unfor-tunately, this choice, which completely eliminatesneoclassical transverse transport and secondary lon-gitudinal plasma currents limiting the equilibrium β,can only be achieved in the vicinity of a straightmagnetic axis. The choice f = f(ρ), known as thequasi-symmetry condition in traps with rotationaltransform, provides new stellarators with tokamak-like confinement (see, for instance, Ref. [11]).

The closed field line systems allow the choicef = f(ρ, λ), where λ is the field line label. Suchsystems have equal length of all field lines on themagnetic surface and are called isometric traps [5].Generally isometry in 3-D systems can exist only ina certain spatial region.1 The pseudo-symmetry con-dition implies merely that the function f should bebounded. In this case the superbanana drift orbitsare eliminated [6]. The pseudo-symmetry conditioncan exist over the whole confinement region.

1 Symmetric 2-D systems are isometric in all space.

The physical sense of the pseudo-symmetry condi-tion (1) can be explained by considering the extremalmagnetic field surface (EMS) [9], i.e. the extremalB value surface along the field lines, B ·∇B = 0.In pseudo-symmetric configurations the condition(B×∇ρ) ·∇B = 0 should be satisfied on all EMSs.This condition indicates that the magnetic config-uration is locally isodynamic, i.e. B = B(ρ) onthe EMSs. In other words, the EMSs intersect theequilibrium magnetic surface along the B = constcontours.

With the appropriate choice of the special mag-netic flux co-ordinates we can reduce the generalcondition (1) to the behaviour of the magnetic fieldstrength on the magnetic surface. For linked mirrorswithout current and rotational transform this is eas-ily seen from the two familiar representations of themagnetic field in the special angular co-ordinates onthe magnetic surface:

2πB =∇Φ×∇θ(3)

2πB = F∇ζ − ν∇ρ.

Here the standard notation is used: F (ρ) is the exter-nal poloidal current, Φ(ρ) is the toroidal magneticflux, ν is a periodic function (ν = 0 in a vacuum),θ = const is the field line label and ζ = const is themagnetic potential surface at ν = 0. Using the firstand second representations of the field in (3) for thedenominator and numerator in (1), we obtain, afterredefinition of the function f , the pseudo-symmetrycondition in the form

∂B

∂θ+ f

∂B

∂ζ= 0. (4)

Since the pseudo-symmetry condition (4) refersonly to a magnetic surface, it is possible to optimizethe magnetic configuration layer by layer. Actually,in order for a plasma to be well confined in a trap,it is necessary to satisfy condition (4) only over anarrow boundary layer. Inside this layer the locallylow transport plays the role of a barrier that acts toreduce the total losses. This possibility is discussedin Ref. [12].

2.1. Isometry in a trap with a plane axisin the paraxial approximation

According to the principle of pseudo-symmetry,the curvilinear elements in linked mirrors should beof the same type as the isolated mirror itself, i.e. ofthe PP type. To demonstrate the existence of suchcurvilinear elements without rotational transform we

946 Nuclear Fusion, Vol. 41, No. 7 (2001)

Article: The EPSILON experimental pseudo-symmetric trap

shall consider the isometry in a trap with a plane axisin the paraxial approximation. In isometric systemsthe lengths of the field line segments on the mag-netic surface between any two B = const contoursare equal. In a vacuum ν = 0 and F = const in (3).Using the standard definitions of the basis vectors

e1 =∂r

∂Φ=√g∇θ ×∇ζ

e2 =∂r

∂θ=√g∇ζ ×∇Φ

e3 =∂r

∂ζ=√g∇Φ×∇θ

e1 =∇Φ =e2 × e3√

g

e2 =∇θ =e3 × e1√

g

e3 =∇ζ =e1 × e2√

g

√g =

∂r

∂Φ·(∂r

∂θ× ∂r

∂ζ

)we obtain the inverted analogues of (3)

2πB =1√g

∂r

∂ζ

(5)

2πB =F√g

∂r

∂Φ× ∂r

∂θ.

We equate the two expressions for B in (5) to obtainthe relation

rζ = FrΦ × rθ. (6)

Here the subscripts refer to the corresponding par-tial derivatives. The relationships between the metrictensor elements gik = ei ·ek can be easily obtained bytaking the scalar product of (6) with different basisvectors:

g23 = g23 = g13 = g13 = 0

g33 =1g33

g22 =1g22

g33

F=√g. (7)

The magnetic field strength can be found by tak-ing the scalar product of the two representations in(5):

B2 =F

4π2√g =F 2

4π2g33. (8)

The isometric magnetic surface r = r(ρ, θ, ζ) isdefined by equations

g13 = 0

g23 = 0

∂g33

∂θ+ f(ρ, θ)

∂g33

∂ζ= 0. (9)

We can seek the desired solution of (9) in the form

x = x0(s0) + sinα · xl + ρ2 cosα · Σ

z = z0(s0) + cosα · xl − ρ2 sinα · Σ

y = yl = b(s0)ρ sin θ

xl = a(s0)ρ(cos θ + ρ∆)

s = s0(ζ) + ρ2Σ.

Here s0 is the axis length, ∆ is the displacement ofthe magnetic surface centres, the co-ordinate systemis associated with a plane curvilinear magnetic axisand the angle α is defined in Fig. 1. The functionΣ = Σ(θ, ζ) is arbitrary. The first two equations in(9) give

Σ = −12

(aa′ cos2 θ + bb′ sin2 θ).

The prime denotes the derivative with respect to s,the other metric tensor elements are

g33 = (1 + 2ρka cosθ + ρ2[−aa′′ cos2 θ − bb′′ sin2 θ

+ 2ka∆ + k2a2 cos2 θ]) · s′20√g = ρabs′0(1 + ρ[2∆ cos θ −∆θ sin θ + ka cos θ])

Here k is the magnetic axis curvature. To the firstorder in the paraxial approximation, ρ� 1, from thelast equation in (7) we obtain

s′0 =ρF

Φ′ab

(10)

∆ =ka

2+ α1 sin2 θ.

Nuclear Fusion, Vol. 41, No. 7 (2001) 947

V.V. Arsenin et al.

Figure 1. Co-ordinate system in the Y = 0 plane.

Here α1 is triangularity. Taking, in the last equationin (9), the arbitrary function f in the form

f(ρ, θ) = C̃1ρ sin θ + C̃2ρ2 sin θ cos θ

we obtain the isometry conditions in first and secondorders:

ka = C1(ab)′ (11)

aa′′ − bb′′ = −C21

2(a2b2)′′ − (2C1α1 + C2)(ab)′. (12)

The choice C1 = C2 = 0 gives the known orthogo-nality condition for 3-D open mirrors. At C1 6= 0 therelation for axis curvature (11) coincides with theformula obtained in Ref. [5]:

k = constB′0

B3/20

exp(η

2

). (13)

Here B0 is the magnetic field on the axis; the mag-netic surface ellipticity is ε = tanh η. If we chooseany function instead of const in relation (13), pseudo-symmetry can be obtained. The change of curvaturesign, i.e. zigzags of the magnetic axis, is the basicgeometrical feature of PP type curvilinear configu-rations. In Fig. 2 the PP curvilinear element calcu-lated by paraxial formulas is shown. The intensityof the isolines shows the B value. One can see thatall B = const lines encircle the magnetic axis. Thereexist solutions to (11) and (12) that lead to toroidallyclosed systems of the bumpy torus type.

3. MHD stabilization by a magneticdivertor of a trap withclosed field lines

Traps with closed field lines are characterized bylarge longitudinal magnetic field gradients. Under

Figure 2. Poloidal pseudo-symmetrical curvilinear ele-

ment. The intensity of the isolines shows the B = const

contours on the equilibrium magnetic surface.

these conditions the realization of the traditionalscheme of MHD stabilization by means of aver-age minB appears difficult if pseudo-symmetry andlarge values of β are to exist [13]. However, trapswith closed field lines have additional, though prac-tically unexplored, opportunities for stabilization. Itis known that for MHD flute-like stability in the caseof closed field lines the condition

−∇p ·∇U +γp (∇U)2

|U | ≥ 0 (14)

should be satisfied [14, 15]. Here p is pressure andγ = 5/3 is the adiabatic exponent.

If the U value has a minimum on some mag-netic surface, stability can obviously occur when theplasma pressure is concentrated about this surface(the first term in (14) is positive). In a trap with atoroidal divertor, U has a minimum on the separa-trix (U → −∞) owing to the formation of a ring cuspwith B = 0. Of practical interest is the situation withthe maximum pressure on the axis instead of on theseparatrix. MHD stability is possible in this case, too,but it is due to the second term in (14), i.e. due tothe large curvature of the field lines. In the languageof hydrodynamics this means taking into account theplasma compressibility. The marginal pressure profileis defined by the equality in (14) and is characterizedby zero pressure on the separatrix [16]

p = const1(∮

dl/B)γ . (15)

Condition (14) is valid for a plasma with finite pres-sure, i.e. the U value in (2) is calculated using thereal equilibrium magnetic field. Thus, if there is anequilibrium with the pressure profile in (15), it is sta-ble. This stabilization mechanism works owing to the

948 Nuclear Fusion, Vol. 41, No. 7 (2001)

Article: The EPSILON experimental pseudo-symmetric trap

0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

2

31

p(r)

/p0

r (m)

Figure 3. Marginally stable pressure profiles, B = 0 at

r = 0.62 m. 1: p⊥ � p‖ (Rp⊥dl is shown); 2: p⊥ = p‖;

3: loss cone energy distribution function.

‘coarse’ effect of non-paraxiality. We do not considerthe possible additional stabilizing effect of the diver-tor, connected with short-circuiting along the B = 0line of oscillating electric fields [17] (tested in theTara tandem mirror experiment [18]).

The stabilizing effect of non-paraxiality also worksin axisymmetric open traps. Such simple traps withdivertors can form the components of a closed sys-tem, providing stability without infringement ofpoloidal pseudo-symmetry. The study of plasma sta-bility in the experimental installation One Mir-ror Element (OME) with divertor is an importantpart of the programme of the Experimental Pseudo-Symmetric Closed Trap (EPSILON). In an isolatedmirror trap the pressure is anisotropic, p⊥ > p‖.The anisotropy strengthens the stabilizing effect ofnon-paraxiality, since a large fraction of the parti-cles are located in the area with strong magneticfield gradient (Fig. 3). As the mirror ratio risesin the area near the separatrix, an isotropic pres-sure model becomes justified. The kinetic consider-ation of stability of Kruskal and Obermann showsa more optimistic pressure profile than (15) [19]. InFig. 3 the calculated marginal pressure profiles forEPSILON-OME experimental conditions with vari-ous assumptions on energy distribution function areshown. Curves 1 and 2 show the marginal pressureprofiles for strong anisotropy and for isotropy of theplasma. Curve 3 shows the intermediate case, whenthe loss cone Maxwellian energy distribution func-tion is realized.

Curves 1 and 3 in Fig. 3 have a higher pres-sure gradient than curve 2 between the axis andr ≈ 0.4 m. However, they have a lower pressuregradient in the outer plasma, between r ≈ 0.4 and0.6 m. The important question of the influence ofpressure gradient reduction near the X point on the

plasma stability will be investigated experimentallyon EPSILON-OME .

According to the results of Ref. [20], all MHD per-turbations in an axisymmetric trap with isotropicpressure are stable if condition (14) is satisfied andthe 1-D equation describing flute and ballooningperturbations without compressibility has not morethan one negative characteristic value. The prelimi-nary calculations show that the limiting β increaseswith an increase of pressure on the separatrix. Forthe experimental study of this effect on EPSILON-OME, an additional source of plasma in the divertorring cusp is provided.

4. The EPSILON experimentalpseudo-symmetric trap

Within the framework of the APEX project atrap with closed field lines and with a plane axis,EPSILON, is examined conceptually. This trap con-sists of two axisymmetric mirrors with divertors andtwo PP type curvilinear elements. The divertorsensure MHD stability and the existence of omnige-neous equilibrium magnetic surfaces. Fulfilment ofthe principle of pseudo-symmetry eliminates largesuperbanana losses.

The equilibrium determines the maximum β

value. Let us compare the known relations for equi-librium β in a tokamak, β ∼ r/Rq2, and in a 3-Dambipolar open trap, β ∼ La/L. Here r is the plasmaradius, R is the radius of the torus, q is the safetyfactor, La is the anchor length and L is the centralsolenoid length. In both systems β is proportional tothe ratio of the minimum characteristic length overwhich B changes (in tokamaks r, in mirrors La) tothe field line length on the B ∼ const section (intokamaks 2πRq ∼ Rq2, in mirrors L). The physi-cal reasons for such equilibrium β behaviour in anymagnetic system are identical. The behaviour is con-nected with the appearance of secondary plasma cur-rents directed along the field lines and caused by thenon-solenoidal character of the diamagnetic currentsin 3-D magnetic systems.

To obtain large β we have two possibilities: usinga rippled magnetic field everywhere (as in the ELMOBumpy Torus [2]) or only in special curvilinear ele-ments with the internally short-circuited secondarycurrents (as in DRACON). In EPSILON the secondpossibility is examined. Closing of the field lines inEPSILON is achieved by making the magnetic sys-tem antisymmetric about the median plane, which

Nuclear Fusion, Vol. 41, No. 7 (2001) 949

V.V. Arsenin et al.

divides the torus into two equal parts [21]. In thiscase the secondary currents are equal to zero onthe median plane. Since the magnetic field is per-pendicular to the mirror symmetry plane x = 0,B = f∇x, the equality B ·∇×B = 0 is valid, i.e.j‖ = j ·B/B = 0. The mirror symmetry and isom-etry can be used for closing the secondary currentsinside the PP type curvilinear elements.

Only a general outline of the system examined isgiven above. It is necessary to perform several tasks:the synthesis of coils for PP type pseudo-symmetriccurvilinear elements with short-circuited plasma cur-rents; large β calculation of 3-D equilibrium in arippled magnetic field; analysis of the topologicalstability of the magnetic surfaces of a configurationwith closed field lines; drift trajectory calculationsand analysis of losses; study of the physics of self-consistent marginal pressure profiles, etc.

5. The EPSILON-OME experiment

Because each cell of the EPSILON trap possessesmirror confinement, it is possible to begin with theexperimental study of separate cells. The experimenton verification of MHD stabilization of the isolatedmirror device EPSILON-OME by a divertor has beenchosen as the first step of the APEX experimen-tal programme. The main parameters of EPSILON-OME are given in Table 1.

The experimental parameters were determined bythe requirement that they be representative and thatthe results obtained would provide a basis for con-fidence in the new method. They were also basedon the need to make maximum use of the avail-able equipment. Meeting the former requirement isensured by having steady state operating conditions,the achievement of high β in a sufficiently large vol-ume and the possibility of verifying the new schemeof MHD stabilization. The available equipment hasdetermined the range of magnetic field and plasmaparameters. A stationary hot electron plasma willbe produced by the large system of 7 GHz klystronswith ECR at two cyclotron harmonics.

The twin divertor coils are separated to obtainbetter access of ECR power to the centre of the trap.Such a configuration also permits creation of a ringcusp trap on the separatrix near B = 0. Plasma pro-duction in this small trap will have an additionalstabilizing effect of the divertor.

The magnetic field configuration of the EPSILON-OME installation is shown in Fig. 4. The posi-tioning of the coils was optimized for better MHD

Table 1. Main parameters of EPSILON-OME

Magnetic field at the centre 0.28 T

Mirror ratio on the axis 2.8

Magnetic field in ring cusp slots 0.28 T

Distance between mirrors 2.04 m

B = 0 radius 0.62 m

Volume of confined plasma 1.25 m3

Klystron frequency 7 GHz

Microwave power 0.5 MW

Plasma density (1–2)× 1018 m−3

Average electron energy 4–20 keV

Confinement time ∼0.003 s

Maximum β 15%

Figure 4. EPSILON-OME magnetic configuration.

1: mirror coils; 2: divertor coils with opposite current;

3: field lines; 4: fundamental ECR surfaces; 5: second har-

monic ECR surfaces.

stability. As the governing parameter we have usedthe gradient of isotropic plasma pressure near theseparatrix in the case of β � 1. The calculatedmarginally stable profiles are shown in Fig. 3. Itcan be seen from the figure that the profiles aremonotonic and are as usual for experimental prac-tice. Pressure anisotropy results in ‘beautiful’ profileswith a flat gradient near the separatrix.

5.1. ECR heating at weak magnetic field

A weak magnetic field is characteristic of plasmasystems with high plasma pressure. For a rather weakmagnetic field the plasma density exceeds the den-sity value determined by the condition ωpe = ω.

950 Nuclear Fusion, Vol. 41, No. 7 (2001)

Article: The EPSILON experimental pseudo-symmetric trap

The planned density indicated in Table 1 is greaterthan the critical value, so the question arises ofthe possibility of producing such a plasma with theuse of ECR in EPSILON-OME. The analysis ofelectromagnetic wave penetration through the denseplasma to the ECR surface was performed inRefs [22, 23]. The main results of the analysis, takinginto consideration the plasma inhomogeneity, are asfollows.

(1) Electromagnetic waves penetrate the criticalsurface ωpe = ω freely if they approach italong the field lines and the angle betweenthe plasma density gradient and the mag-netic field does not exceed the value χc =arccos

√(ωce − ω)/(ωce + ω).

(2) In open systems there is the possibility oflaunching EC power through the mirrors at asmall angle to the axis. The value of the admis-sible angle is strongly influenced by the raytrajectories’ instability due to refraction in aplasma whose density decreases away from theaxis. Calculation of the transmission coefficientof the wave beam for conditions close to thoseof the EPSILON-OME experiment shows thatthe launching angle must not exceed 10◦.

Two systems of ECR plasma production aredesigned for the EPSILON-OME experiment. Thefirst system has longitudinal near-axis microwaveinput for dense plasma production. The second sys-tem has transverse input located in the divertor forplasma production in the cusp region.

6. Conclusion

A solution to the development of a stationarymagnetic trap capable of high β plasma confinementwith a tokamak level of losses may be found in theconvergence of closed and open systems. Ideas ofpoloidal pseudo-symmetry and divertor MHD stabi-lization offer a new approach to the old idea of mir-rors linking into a closed system. Systems with closedfield lines are poorly studied both theoretically andexperimentally. Within the framework of the APEXconceptual project we are going to study the problemnot only theoretically but also experimentally. Thefirst step of experimental activity is the constructionof the EPSILON-OME installation. Verification ofdivertor MHD stabilization is the basic point of theproject. The start of experiments is expected withintwo years.

Acknowledgements

Numerous discussions with V.D. Shafranov hada determining influence on our study. This work ispartially supported by the Russian Foundation forBasic Research, contracts 00-02-17105 and 00-15-96526, under its programme of support to scientificschools.

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(Manuscript received 4 October 2000Final manuscript accepted 19 March 2001)

E-mail address of A. Skovoroda:skovorod@nfi.kiae.ru

Subject classification: B0, Mt; E0, Md

952 Nuclear Fusion, Vol. 41, No. 7 (2001)