Ming, L.X.

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A Combined Scheme of Adiabatic Compressionand Accelerated Merging of Tokamak ToriMing-Lun Xue*Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139PFC/RR-81-28July 1981*Visiting Scientist, on leave of absence from the Institute of Mechanics, Chinese Academy of Sciences, Beijing, ChinaA Combinert Scheme of Adiabatic Compressionand Accelerated Merging of Tokamak ToriMing-Lun Xue*Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139AbstractIt is shown that inclusion in the analysis of the surface current induced during rapid adiabatic compressionresolves the main discrepancy between the simple scaling laws of Furth-Yoshikawa and detailed numericalcalculations. A scheme combining adiabatic compression and accelerated merging, producing no net surfacecurrent, is suggested.*Visiting Scientist, on leave of absence from the Institute of Mechanics, Chinese Academy of Sciences, Beijing, China.I- - - 1,1. IntroductionAdiabatic compression along the major radius is a simple way to increase the temperature of tokamakplasmas. However, from an economic point of view, this heating method is not likely to be viable in futurefusion power plants because of its pulse type work in nature and, after being compressed to a much smallermajor radius, only a small portion of the toroidal magnetic energy can be utilized. Nevertheless, from theexperimental point of view, adiabatic compression may be an inexpensive way to reach the temperature thatmust be attained in the experimental ignition facilities.Furth and Yoshikawa' derived some simple laws for adiabatic compression several years ago. Recently,however, more detailed calculations have shown that large discrepencies exist between numerically calculatedtoroidal current and poloidal beta 2, 3 4 and those predicted by the Furth-Yoshikawa scaling laws. It is one of thepurposes of this paper to attempt to resolve the discrepancies. Some authors2,3 have claimed that the differenceis due to the assumption of large aspect ratio in the Furth-Yoshikawa theory. However, we will show that thediscrepanices can be resolved unambiguously if we take into account the induced surface current which wasneglected in the analysis of Furth and Yoshikawa. In this regard, it will also be shown that the assumptionof large aspect ratio plays an insignificant role. It will be shown in this paper that the direction of the surfacetoroidal current is in the same direction as that of the main toroidal current, this surface current may degradethe stability properties of toroidal plasmas. Albert3 suggested regulating the poloidal flux of the Ohmic heatingcoils accordingly to compensate for the surface current. It is doubtful that this method could be used practicallysince the time constant of the Ohmic heating coils is much larger than that of the vertical field coils.Recently another method 5'6 to increase the plasma temperature has been suggested. Principally, there isan electromagnetic attraction force between two separated tori, with the toroidal currents in the same direction.During the acceleration process, part of the poloidal magnetic energy is converted to the kinetic energy ofthe plasmas and is subsequently dissipated as thermal energy by collision. Recent experiments have givenencouraging results. In comparison with adiabatic compression, this method allows simplification of the designand arrangement of the toroidal coils. One possible problem that might exist is that a surface current in thedirection opposite to that of the main current is also induced during the accelerating period; this may alsodegrade. the stability properties of plasmas.A new scheme of heating is suggested, combining adiabatic compression along the major radius andaxial acceleration of the tori. Since the induced surface currents are in the opposite direction when thesetwo processes work separately, it is possible that the surface current can be cancelled when they work simul-taneously under certain conditions ,without regulating the Ohmic heating coils.2"MR9In order to get a clear physical picture of this process, we analyze pure radial adiabatic compression andpure axial acceleration. We then consider the combined scheme.II. Pure Radial Adiabatic CompressionEquilibrium of toroidal plasma needs an applied vertical field, and the total poloidal magnetic field is thecombination of the induced poloidal field of the plasma and the vertical field. In a low beta plasma,.the inducedtoroidal field is much smaller than the applied toroidal field, so it can be neglected without introducing muchinaccuracy.If a surface toroidal current is induced in an infinite conducting plasma, it does not change the magneticstructure inside the plasma directly, but it does change the poloidal magnetic field outside the plasma torus.Since the total toroidal current has a step change across the plasma boundary, the , and q also have a stepchange across the boundary. Now we can divide the total toroidal plasma current into two parts: one is thevolume current I, and the other is the surface current It,During the adiabatic compression process of ideal plasma, two conditions must be satisfied within the plasma:a2Bt = const. (2)q(a-) = const. (q(V) =const.). (3)a+, - means the minor radius just above or below the plasma surface. From these two conservation conditions,we can derive the Furth-Yoshikawa scaling laws (with BtR = const.).from(2) Booa- 2  (4)a_.Bj a._.Btfrom(3) .q(a) = 00 = const. (5)RB,(a) RIt,1afrom(5) Itoo1 (6)the last equation shows that the volume toroidal current is inversely proportional to the major radius R. But (2)and (3) are not the only two conditions that must be satisfied during adiabatic compression; the total poloidalmagnetic flux outside the torus must also be satisfied during adiabatic compression.4ir R f~Oout = -R(In8 - 2)It - R B_27rrdr = const. (7)C a fo3.nd from the equilibrium conditionBL = (R n8 a+# -P + )(8)Here we treated two limiting cases:Case I: B± is homgeneous in the whole regionthen Eq. (7) can be integrated to47r R tI(n8R 3 1ipout = R(1n8 _ 2)It - (n8+ + )7r(R - a)2  const. (9)soI42 R, [ln8Ri/ai - 2 - 1/4(ln8Rj/aj - 3/2 + #,i(a+) + 4i1/2)(1 - ai/R) 2]It . R 2 [ln8R2/a 2 - 2 - 1/4(ln8R2/a 2 - 3/2 + 1p2(a+) + l2/2)(1 - a2/R 2 )2]where(p)Op(a+) = PB2(a+)/81rCase II: B_ = 0 outside the plasma41rtRl R'Pout tR(n8 -- 2) = const. (11)I2 RI ln8Rj/a1-2It, - 2 in8R2/a 2 - 2The difference between I42 and I, is the total surface current It.,t,= IA2 - tv,.It is clear from the above analysis that Furth-Yoshikawa had only considered the I, without considering theexistence of It, 2. It is also shown that the q(a-) is conserved during the rapid adiabatic compression, but q(a+)is not conserved. Since a+ is already in the vacuum -region, there is no reason why q(a+) must be conserved.The real case is between these two limiting cases. It does show a large deviation from the Furth-Yoshikawascaling law on the total toroidal current and poloidal beta due to the existence of surface current whether or notthe aspect ratio is large enough (Fig. 1).4III. Pure Axial Acceleration and MergingThe total magnetic energy of two mutually interacting tori isW = 2( 1 L ) + I1 (12)where L, It is the inductance and toroidal current of one torus, and M is the mutual inductence.The force along the Z direction isOW IlOMF,= law, t(13)and the equation of momentum isdV M dV I (OMdt- = MV = C2 (14where Mp = 2 X (n)mgira2 X 2irRi since magnetic flux is conserved during acceleration. That is,(L + M)I = const.. (15)Substituting (15) into (14) and. integrating with respect to zv2  12ML =M2L+M (L + M)2(L (16)As a rough estimate, we assumed M, P 0 for two initially well-separated tori and M2 j L when the two toriare next to each other. Then from (16)V2 122t (17)For example, two well-separated tori with parameters near Alcator A, the final velocity isV2 _ (12 )1/2withMp = 2 X (n)mjira2 x 27rR= 2 x 2 x 1014 x 1.67 x 10-24 x 7r(9.5)2 x 27r x 54= 6.4 X 10-~5gI = 101A5V2 P 0.54 X 108cm/sec.and.h2 +A Ai (if < ) (18)since q(a....) is conserved, so the volume current is conserved. If the total current decreases to half of its initialvalue, then a surface current-must be induced in the opposite direction of the main electric current. The changeof the poloidal magnetic energy during the acceleration process isAW = 2( 12) +MI i- LIt2 (19)1 L1(Iand it is consistent with the expression (18). The total initial thermal energy isWth = 127r2a2RnkT. (20sothatA Wp 1 4 R -Wi Op 6 a i ,Axial acceleration and merging seem to be an efficient heating method only for low p, and not for high p,tokamak. The poloidal magnetic energy that can be released to heat the plasma is quite limited.IV. Combined Scheme of Radial Compression and Axial AccelerationSince pure radial compression and pure axial acceleration induce the surface currents in the opposite direc-tion, it is natural to consider combining these two processes. Under certain coditions, it is possible to obatin adynamic process without inducing surface currents.Fig. (2) illustrates two plasma tori being compressed and accelerated along paths making an angle a withthe horizontal direction. The following conditions are still satisfied within the plasma.Bia2 = const. (22a)BtR = const. (22b)q(a.) = const.. (22c)6From these three conditions, we obtainacoR"/2  (23a)Itvo . (23b)Although it is not difficult to introduce any distribution of the vertical field outside the plasma, we consider herea simple case where the vertical field is zero outside the plasma.Another condition that must be conversed isTout = (L + M)I = const.. (24a)That is,dI d(L+M) (24b)hL+M(4)from (23b) we obtaindIv -dR (24c)'tv -since what we need is It = 0, It =Iv; that is to say, no surface current is induced, sodIt- dltv (25)It Itvput (24a), (23c) and (25) togetherdR dL + dM 26)R L+M L+M(26dL OM/ORdR + MOzdzL+M L+M= dz L + MIR - dL/dR -aMORg= = c9M/Ozwhere7RL = 47rR(ln8- - 2)aM = [(1 - k2)K - ElR 2 + z2K= fn/2 dO2K' 1-k 2 sinGIr/2E= V1 - k2 sin2 OdO9M 2rR-- = [K-El'5 V/R 2+ Z2OM 47rz R 2 +2z 22 - = -[-K+ 2-2 E]z R2+Z2zThe real path is the simple numerical integration of Eq. (26), as an initial value problem, for illustration wechoose a typical value (z/R = 0.5) to estimate the local angle aa|z/R=0.5 = tg'1 1.03 P 460V. Conclusions1. Rapid major radius adiabatic compression induces a surface current on the torus, which Furth-Yoshikawapreviously neglected. It is this surface current, and not the assumption of large aspect ratios, that causesthe main discrepancy between the simple scaling laws and more detailed numerical calculations.2. Axial acceleration with merging does not seem to be a convenient heating method for high Op (3p, > 1)plasmas.3. A combined scheme of radial compression and axial acceleration is suggested. The surface current couldbe completely cancelled under certain conditions. The typical compression angle is a s 45'.8REFERENCES1. Furth, H.P., and Yoshikawa, S. 1970, Physics of Fluids, 13, 2593.2. Holmes, J.A., Peng, Y-K. M., and Lynch, S.J., 1980, Physics ofFluids, 23, 1874.3. Albert, D.1, 1980, Nuclear Fusion, 20, 939.4. Holmes, J.A., Peng, Y-K. M., and Lynch, S.J., 1980, J. ofComputational Physics, 3, 35.5. Shafranov, V.D., 1979, Nuclear Fusion, 12, 187.6. Ohyabu, N, Hsieh, C.L., and Jensen, T.H., 1978, J. of Plasma Physics, 21, 253.7. Landau, L.D. and Lifshitz, Z.M., Electrodynamics of Continuous Media, Pergamon Press, 1960.8. Abramowitz, M. and I.A. Stegan, Handbook of MathematicalFunctions, Dover Publications, Inc., 1965.9FIGURE CAPTIONSFig.1. The variation of total toroidal current (a) and poloidal beta (b) as compared with simple lawspredicted by Furth-Yoshikawa's theory. Case I: Vertical field is homogenous. Case II: Verticalfield is absent outside the torus.Fig. 2. Combined Scheme of Radial compression and Axial Acceleration without inducing surface current.10ItIt,21= 1.7at42p I(b)FIG. I6I 1.7R,= 4cosets2--- .tV2It,5431RR2a2(F-Y)R(R,(F-Y)3(a)3-I42!-03 4C 0 S- e31I(R/g'VARRFIG. 2

Combined scheme of adiabatic compression and accelerated merging of tokamak tori

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Combined scheme of adiabatic compression and accelerated merging of tokamak tori

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