p. c. t. van der laan and l. w. mann- tokamak equilibria with beta close to 1

8/3/2019 P. C. T. Van der Laan and L. W. Mann- Tokamak Equilibria with Beta Close to 1

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TITLE: TOKAMAK EQUILIBRIA WITH f4CLOSE TO 1

%-+. C. T. Va r Laan and L. W. Mann

SUBMITTE D TO: p Roc~E DINGsOF THEHIGH-BETAHEORYo~sfiopVARENNA CONFERENCE, September 1977.Varenna, Como (Italy)

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TOKAMAK EQUILIBRIA WITH ~ CLOSE TO 1

P. C. T. Van der Laan and L. W. MannLos Alamos Scientific Laboratoryof the University of C7.iforniaLos Alamos, New Mexico U.S.A.

Equilibrium calculations have been done for a High-Beta Tokamakexperiment, that has been under consideratio~.in Los Alamos.1 The geometryztudied was that of a torus with a uajor radius R. of 30 cm and aracetrack-shapedminor cross section of a height of 48 cm and a width of24 cm.A uetal shell around the discharge vessel keeps the magnetic surfaces olosedand inside the vessel so that force-free currents can flow along the fieldlines. Results showing equilibria at very high beta have been obtained with acomputer program described in Ref. 2. It appears that the beta limit forequilibrium has disappeared and thst the force-free currents and the elongationof the minor cross section allow equilibria with a beta close to 1 at a smallshift of the magnetic axis. There is a stron& relation between thesepinch-like equilibriaand the Flux Conserving Tokamak (F.C.T,).

earlier for a possiblequilibrium studies done High-Beta Tokamakexperiment havtibeen extended to higher betas. The incentive for these studieswas!the Idea that these equilibria should be possible if no separatrix isallowed to come inside the plasma,3-5 in other uord~ if q AS kept finite. Inadditicn to this condition on q, the two trial functions p(w) and I(GJ)used Inthe computer program we r e chosen to approximate a flat q(r) profile. There are


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,.

three reasons for this choice: theflat q profile reduce the otherwise

-z-force-free currents corresponding to thisexcessively large equilibrium shift of the

plasma, secondly a flat q profile gives the best chance for stability6 andthirdly flat q-profiles are easy to generate experimentally, as has been shownin Screw Pinches. The two functions that thus far have produced the bestequilibria are

{P = P. x - 0.6(1 -X)3 sin~ } forx>O

() 1/2lJoI= ROBO 1 - 2C : + a$o(1)

(2)

where x = W-~)/(4Jm-lJc),$ is the poloidal flux times l/2?T,which runs fromzero at the wall to Wc where the pressure starts to rise, to its maximum $m at

the magnetic axis. The plasma pressure, p, has its maximum p. on the magneticaxis. The poloidal current times l/2n is denoted as I, R is the distance tothe major axis, R. is the major radius and B. is the toroiddl field at radiusR.. Equation (1) gives p an almost linear dependence on $, wihh a rounding offnear the low-pressure edge. The first term on the right hand side of Eq. (2)generates the diamagnetic dip in the toroidal magnetic field with a pressuredependence as in a high-beta theta pinch. The last term in Eq, (2) representsthe force-free aurrents. The coefficient ~ is chosen to give a reasonably flatq(r). The toroldal current density on the magne axis, j,,% required to

b


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d

, . . .,,

-3-produce the right q value on axis, can be used to find the coefficient C bymeans of the equation

(3 )

evaluated on the magnetic axis with Eqs. (1) and (2).One of the many equilibria that have been found is described in Table I

and Figs. 1 and 2.

TABLE IPARAMETERS FOR EXAMPLE OF EQUILIBRIUM

INPUTpo(MPa)m

60.540 0.:;3

0= BO(T) c %->0.3 ymax 2.0 0.44

OUTPUT@rms Y&l Shift (cm) Lf(nH)9.713 L 4.46 119.1

The three 8 values ligted in the table are: the averafle toroidal 6,2 P.

/Bo2 often used in Tokamak literature; the local 6 on the magnetioaxis, 2 popo {2 lJOPO }+ 6$2(rx) l; and a root mean square 6, defined as

r]Jo/Bo2 which should be a figure of uerit for fusion reaotorextrapolations. The inductance Lf is the inductance 27TlJm/10 which theIcapacitor bank sees.


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. . . .

-4-The peak pressure and the ~ values for this example are very high. It is

~l!nostcertain that equilibria with @ax = 1 exist also, although no se~iousattempt has been made to find out what p(~) and 1(0) functions would be neededfor this rather degenerate case. A reasonable conclusion is that the $ limitfor the equilibrium has disappeared. This is probably a general conclusion forflux conserving equilibria but the elongation cf the cross section and theforce-free currents are required to reduce the outward shift to a tolerablevalue, such as obtained her..

Figures 1 and 2 illustrate the equilibrium of Table I. Figure 1 shows atthe left the flux plot in the top half of the racetrack and at the right thedistribution of the toroidal and the pololdal field, the pressure and thesafety factor q along the midplane of the torus. The value of q is greaterthan 1 everywhere. At these high 6 values an elongation of the innermostuagnetic surfaces takes place, which increases the q values near the uagneticaxis. Either this Increase of q or the elongation, which increases the chancefor ballooning, lead to a violation of the Mercier criterion near the magneticaxis. How relevant this is, or whether this could be rectified bymodifications of the p and I functions, or by flattening of the p-profile in anactual experiment is an Gpen question.

Figure 2 shows the distribution of the current densities along themldplane of the torus. At the left the poloidal and toroldal current densitiesare plotted: nc?e the large jpol, necessary at this high ~. At the right theperpendicular and parallel currents are shown.


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-5 -The parallel currents flow throughout the plasma, also in the pressureless

plasma outside the pressure profile. These force-free currents contributestrongly to the toroidal current and reduce the toroidal shift of the plasma.lheerpendicular currents flow Inside the pressure profile only and are peakedat the cutside, similar to what is seen in the Oak Ridge FCT.5

High-beta equilibria, such as described could be produced rather easily inScrew PZ~ch-like uachines. Required are the closely fitting shell, a uodestvertical elon~ation and a programming of external currents to ge~erate the flatq-profile. Heating is in fact easier for these higher betas because theimplosion heating is uore efficient at a lower bias field.

The crucial question for these high-beta Tokamak equilibria is what the ~limit for stak.itywill be. Existing stability theories6 show the beneficialeffect of both the elongation and the force-free currents, but are not directlyapplicable because the model used has a sharp skin plasma, has no wallstabilization and has separatrixes present in sc}mecases. An extension of theequilibrium code used, can test for MHD stability, but there are problems withthe required numerical accuracy.

Experimental evidence for at least a fair zmount of stablq.itycan beinferred from Screw Pinch7 and Belt Pincha experiments in which lifetimes ofa out 200 us were obtained. The problem in these experiments is that theinitial f3 of respectively 20 and 60Z decays, so that a conclusion aboutlong-term stability id not possible. To avoid this decay one nee~s cleanersystems and operation at higher temperatures, so that especially the Intenseradiation of Lithium-like ions, such as O VI, can be avoided. The strong


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I

-6-Initial oscillation in the midplane around the eccentric equilibrium position

proves that the field configuration is able to hold both plasma pressure andoutward momentum. This provides evidence that higher B values could be heldalso, but does not say much about stability because of the short duration ofthe maximum outward excursion.

Although both theory and experjrnentprovide these promising indications,no clear information on the stabiliky ~ limit Is available at this stage.

Three theoretical studies, possible with the ~aker-~ann code are brieflydescribed here.

. The parameter $C In Table I can be increased to produce a narrower plasmaprofile. Results obtained thus far, show that the elongation of themagnetic surfaces can be retained even when the profile width at half POis reduced to 8 cm. These studies can give information on thea~cessibility of elongated equilibria at higher compression ratios, (aproblem discussed in Ref. 9). Results quoted in Ref. 10 show that theplasma has less tendency to shrink to a circular cross section Ifforce-free currents are present.

q The stability might be imprGVed if the uinor cross section of the machineis given a D-shape. To facilitate construction we chose a shape in whicha circular bulge was added to the outside of the racetrack, and the insideof the racetrack was kept straight. A number of equilibria in thisgeometry hes been studied; a conclusion is that the Mercier unste.bleregion near the uagnetic axis shrinks; however the peak pressure in thesestudies was only 1.5 MPa.

.


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q The force-free-7-

currents C1OS,?to the wall can be made to fall off by areduction ofindicates that

near the wall. Experimentalll and theoretica1126evidencethis has little effect on equilibrium and stability. It IS

nevertheless interesting to study this effect in uore detail, because thecurrent channel is expected to shrink at later times.Both in the FCT and

done, flux conservationconserved from the uoment

in the HBT uachine, for which the calculations wereis of crucial importance. In the FCT fluxes arewhen powerful additional heating is applied to the

conventional tokamak target plasma. In implosion-heated Takamaks, theconfiguration is set up and an important part of the heating takes place duringthe implosion. Eecause a large heating power (order GW) is fed into the plasmaduring this setting-up phase (order I-Is)fluxes are conserved right from thebeginning. This conclusion could be valid for pinches in general, but thereare two restrictions.

q Pinches operated atconsequently large

lower densities induce large drift velocities and haveanomalous resistivities. This provides an efficient

ueans of heating but leads also to an enhanced interdiffusion of field andplasma and hence to poor flux conservation.

q Pinches constructed without a conducting shell or with an externallyapplied vertical field can only be flux conservin~ in that limited part ofthe cross section where the flux surfaces remain in a well conductingplasma.


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I

-8-In Screw Pinches and in the liBT-plans,the ~onstruccion of the shell and

tilechosen density range are such that both restriction: are largely avoided.This leads tc a cmservation of the safety factor q for an observer uoving wi.hthe field lines, in other words: Dq/Dt = O. This Is the same condition as theone used in FCT desoriptims where q(~] is constant.

The initial q(o) profile which stays constant during the heating phase inthe FCT, is that of the conventional Tokamak discharge, which is formed beforethe additional heating starts. Curve a in Fig. 3 shows such a q profile.

The q profile in Screw Pinches and in the HBT 1? built up during theformation, when the field lines move in. Since the field lines carry In theirq value, the time history of q at the wall determines the resulting spatial qvaliation. Curve c in Fig. 3 represents the result for the simples~programming: a q(wall) constant in time produces a uniform q in space. Togenerate curves a or b, q(wall) should rise in time. Such a time behavior ofq(wall) can be obtained by slowing down the rise of the plasma current relativeto the rise of the toroidal field. Appropriate modifications in t;~eexternalcircuits make it.in fact possible to generate any given q-profile. Profilessuch as a have the force-free currents that are also present in regular Tokamakdischarges; a profile as c has additional programmed folce-freecurrents.

REFERENCES1. P. C. T. Van der Laan, J. P. Freidberg and K. S. Thomas, ProPO:al for the

Construction of a High-E?etaTokam~k at LASL, Los Alamos report LA-6413-P1976.


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-9-2. D. A. Eaker and L. W. Harm, llEquilibri.?tudies for a High-Eeta Tokamak~

Bull. Am. Phys. SOC. Q, 642 (1974). Also D. A. I?akf?rnd L. U. Mann,MIIDStability Studies of Kumerlcally Obtained Toroldal Equilibria, Proc.2nd Top. Conf. on Pulsed HiSh-Eeta Plasmas, Garching report IPP1/127,Paper E7, 69-72 (I?T2).

3= V. S. Mukhovatov and V. D. Shafranov, Plasma Equilibrium in a Tokamak,llNucl. Fusion U, 605-633 (1971).

4. P. C. T. Van der Laan, k. Schuurman, J. K. A. Zwart and J. p. (loedbloed,On the Decay of the Longitudinal Current ia Toroidal Screw Pinches,n IAEAProc. 4th Int. Conf. on Plasma Physics and Contr. Nucl. FusionRe~ear~h, Madison (1971) Vol. 1, pp 217-223.

5. R. A. Dory and Y-K. M.-Peng, High-Pressure, Flux-Conserving TokamakEquilibr!.a,Nucl. Fusicnm, 21-31 (1977).

6. D. A. D~Ippolito, J. P. Freldh>rg, J. P. Goedbloed, and J. Rem,ll~aAimiz~ng B in a Tokamak with Force-Free Currents~ttSixth Intern. Conf.on Plasma Physics and Controlled Nuclear Fusion Research, Berchtesgaden,Federal Republic of Germany, Oct. 6-13, 1976, Vol. I, 523-5~5.

7. A. A. M. Oomens, C. ~beldijk, J. A. Hoekzema, A. F. G. Van der Meer, andDo Oepts, IiInfluenceof PredischargeConditions in SPICA~ proc. 8th Eur.Conf. on Contr. Fusion Bnd Plasma Physics, Prague 1977, p 71.


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8. E. Graffmann,*Experimental

-1o-F. Hoenen, A. Kaleck, L. K&?n, M. Korten, and J. sChl;tFr,investigations of the Stability of a Pelt-Pinch to Vertical

Displacements, Prague, p. 77.

9. A. Kadish and D. C. Stevens, ~Equilibriaand Adiabatic Compression ofFres-EloundaryBelt ?inches,!!Nuc1. Fusion N, 821:829 (1974).

100 J. A. Hoekzema, ltTorojdal Equilibrium of Non-Circular Sharp BoundaryPlasmas Surrounded by Force-Free Fields,lt in &h_lsedHiKh Eeta Plasmas,(Culham Laboratory, Abingdon, September 9-12, 1975) PergarnonPress, NewYork (1976) 535-539.

11. C. Bobeldijk, J. A. Hoekzema, M. Mimura, D. Oepts, and A. A. M. Oomens,IIcurrentDecay and Stability in SPICA,lSixth Intern. Corlf, on PlasmaPhysics and Controlled Nuclear Fusion Research, Berchtesgaden, FederalRepublic of Germany, Oct. 6-I3, 1976, Vol. I, 493-500.

12. J. 0. Hoekzema, llDe~ayand profile of the TorGidal Plasma Current in aScrew Pinch,!tin Pulsed Hi~h Beta Plasmas, (Culhdm Laboratory, Abingdon,September 9-12, 1975) ?ergam>n Press, New York (1976)541-545.


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Fig. 1.

Fig. 2.

Fig. 3.

FIGURE CAPTIONS

Flux plot for the top ha]f of the racetrack and plots of B+Pp, and q &long the midplane of the torus. The safety factor qis calculated for each magnetic surface and is plotted at eitherside of the magnetic axis. Note Lhe relatively small shift ofthe magnetic ax% at this highequilibrium are given in Table

pressure. The parameters for thisI.

Tilecurrent densities for the equilibrium of Table I and ?ig. 1plotted along the midplane of the torus. Inside the pressureprofile which is also shown, both parallel ard psrpendic~lar cur-rents are flowing, vhereas only parallel CULrCnt.S can flow when Phas dropped to zero. The poloidal currents change sign in thatregion, thereby separating the parama~netic outside from thediamagnetic inside

Various q profiles

plasma.

that can be produced in Screw Pinches or inthe hBT. Carve a corresponds to the FCT profile; curve c showsthe flat profile required for the high-beta equilibria discussedin the text.


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p. c. t. van der laan and l. w. mann- tokamak equilibria with beta close to 1

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