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Physics Basis for Mftf-B UCID-184%-Part 2 Lawrence Livermore Laboratory UkMirror Fusion % PHYSICS BASIS FOR MFTF-B Editors: D. E. Baldwin B. G. ' ogan T. C. Simonen January 16, 1980 Tills Is in informal report lutntei prima rill for Interrul or llmltrt atmul ibtrSbnliom, The opiahnis and condmion stated are those of die author wd may or may Mt be those of the Laboratory. Work performed uider the ausptu* pf the U-S. Department of Energy by (tie Lawrence Livermore Laboratory uaoti Contract W-7405-Eng-48. .„B™^Norr^^^NTlsUNL.f«.TO IV. PHYSICS TOPICS IV.A. ADIABATICITY IV-3 R. H. Cohen, '£. A. Cutler, and J. K. Foote IV.B. MHD STABILITY IV-13 T. B. Kaiser IV.C. RADIAL TRANSPORT IV-43 M. E. Rensink, R. H. Cohen, and A. A. Mirin IV.D. BARRIER PHYSICS , IV-79 D. E. Baldwin IV.E. LOSS-CONE INSTABILITY IV-109 L. D. Pearlstein, G. R. Smith, and W. M. Nevins IV.F. EFFECT OF GAS RECYCLING AND SECONDARY ELECTRON EMISSION IV-151 G. D. Porter IV.G. MONTE CARLO SIMULATIONS OF TANDEM MIRROR PHYSICS IV-177 T. D. Rognlien IV.H. CHARGE-EXCHANGE PUMPING OF THERMAL BARRIERS IV-195 G. Hamilton IV.I. VACUUM PUMPING IN THE A-CELL IV-209 G. E. Gryczkowski and G. D. Porter IV.J. MICROSTABILITY OF THE CENTRAL CELL IV-215 G. R. Smith and D. E. Baldwin IV-1 IV.A. ADIABATICITY R. H. Cohen, T. A. Cutler, and J. H. Foote IV-3 . fr IV.A. ADIABATICITT R. H. Cohen, I. A. Cutler, and J. H. Foote The degree to which magnetic moment is conserved for A-cell ion proves to be an important consideration in magnet design. The A-cell is the only component of MFTF-B that required changes in preliminary field designs to satisfy adiabaticity requirements. We discussed adiabaticity in the plugs in the MFTF proposal ; the present plug magnetic field design adiabatically confines 400-keV ions even at 3 • 0.5. Adiabaticity of ions confined in the solenoid is also satisfactory (as discussed at the end of this subsection). We know that particles in mirror fields experience jumps in their mag­ netic moments; these jumps usually occur near the minimum of the field strength B along a field line. For a highly nonaxisymmetric magnetic field, 2 these jumps are almost always stochastic, and hence give rise to pitch- angle diffusion- In the MFTF-B A-cell, 80-keV ions injected partway up the mirror (at mirror ratio R •= 2) fall through a potential drop of about 50 to 100 keV be­ fore reaching the bottom of the magnetic well. The issue is whether the re­ sulting 130 to 180 keV ions nonadiabatically diffuse in pitch angle on a time scale short compared to the desired collisional lifetime (about 1 c). A use- 3 ful measure of adiabaticity is the adiabatic energy limit W ., defined to be the energy at which the nonadiabatic diffusion time is equal to the desired (collisional) lifetime. Because of the otrong dependence [~exp(const * W 1 of the nonadiabatic diffusion time on energy W, an energy limit W . signifi­ cantly below the mid-machine ion energy of 130 to lfiO keV would imply signifi­ cant enhancement of pitch-angle scattering over the collisional rate, and hence signficiantly increase neutral beam and barrier pumping requirements. A-cell adiabaticity was examined using both the TIBRO orbit code and the analytic theory. Since theory predicts that the magnetic moment jumps per bounce Ay should be umall for field lines whose minimum-field radius is smal­ ler than v/Sl (where fi is the minimum gyrofrequency), we concentrate on field lines far from the axis. According to theory, Au is given by AH £K « Max {£- , ll xA £-) exp(-K/e) cos * (1) IV-5 where the subscript o refers to values at the minimum of B on a field line; e " mcv/qB L (the specific choice for L is unimportant so long as it is self-consistent); and A is a weak function of energy and field line chat has a numerical value of about 4. The quantity ic/e scales as BL/i'S, where L is an axial field scale length and ia found as followsr Consider the gyrophase # as a measure of position along a field line for particles of fixed W,u in between one pair of bounces. Analytically, continue the function B(<JJ) into the complex ifi plane end find the set of complex zeroes {i{i.} of B that lie over the portion of a field line between bounce points (i.e., their projection onto the real axis lies in the range of * incurred in a single transit). Identify ip^, the i|i. with smallest imaginary part. Then ;/e *r - Re^. We obtain analytic estimates for Au from the numerically computed design field by least squares fitting the field strength by a polynominal in S, the arc length along the field line. We do not attempt to fit the entire line, but only the portion with mirror ratio less than 3.2, since we are in fact only interested in particle* that turn at mirror rate of about 2. As shown in Fig. IV.A-1, the fitted field is a good approximation to the numerical field in the fitted region. The solid curve in Fig. IV.A-2 shows (Au/u) calculated from Eq. (1) caking A - 4 and cos ifr • 1. The circles in Fig. IV.A-2 denote numerical values obtained from the TIBRO orbit code. The code calculates an average u of the instantaneous 2 u = (mv^/2B) over the last several gyroperiods before each bounce. An orbit is followed for several bounces, and the largest change in v is selected to pick out Au for |cos * | =1. Agreement between TIBRO and analytic results is seen to be quite satisfactory. 3 The adiabatic energy limit tf . is determined using the formula 3 2/M 1,0 x 10" (K BQ L Z) Wad " (l - 0.636 In A ) (2) where 2 15(v 2/v2) <V /V) 1/2 , L -2 v r Hn X1 2" I* M 50 keW *p 0.5 10_l_sec\ f __ o o j *2 W \170 cm <v,|>/v T / | A 0.59 <COBI)I r>J ' (3) IV-6 co 2 -2.0 1.0 2.0 Fig. IV.A-1. Magnetic field strength B vs arc length S for field line and quartic fit. IV-7 lVE<MeV» Fig. IV.A-2. Analytic (solid line) and numerical ((x)'s) values of &W V for field shown in Fig. IV.A-1. IV-8 Here, L is the plasma length (used to calculate a mean transit time L /<v >), which might be chosen as a definition for the field scale length L, ? is the desired lifetime, M is atomic mass, Z the charge state, and <cos Ji > the rms value of the gyrophase (taken here to be 2-1/ 2 ). Here, L is in cm, T is in s, and B is in kO. Determining KBL from our analytic expression for K/E, we find, from Eq. (3), an energy limit of 350 keV for the design studied. Hence, in vacuum, the magnet studied appears to give good adiabatic confinement. The gross scaling properties of the energy limit may be seen from Eq. (2). A change in midplane magnetic field made, holding the peak magnetic field and the distance between mirrors fixed tends to make the energy limit scale 2 3 faster than B ; more typically as B , since the effective scale length L tends to increase with B . A subtler effect is the dependence of the energy limit on the shape of the magnetic well. The field profile shown in Fig. IV.A-1 is fit by a poly- nominal containing all orders of s through quartic. If v-i consider a class of fields that has the same minimum field strength and the same field strength at z = +1-35 m, but vary the ratio of the quadratic and quartic coefficients, we find that there is an optimum ratio between all quartic (steep-edged, flat- bottomed well) and all quadratic (gentle well). In fact the optimum is very close to the shape of Fig. IV.A-1. The energy limit at the optimum is about 360 keV; for no quadratic, it is 200 keV; and for no quartic, it is 220 keV. Finite 8 lowers the energy limit; but because of the off-midplane peak­ ing of the hot ion distribution and the relatively broad pitch-angle distri­ bution of hot electrons, 8 should be not especially peaked at the minimum of B. Hence, in contrast to the situation in charge-exchange-dominated, perpen­ dicular-injection, neutral-'i>sam mirror machines such as 2XIIB or the plug cellB of TMX or HFTF-E, the effect of finite B on the energy limit should be mainly to lower B (as opposed to also producing a dramatic drop of scale ° 1/2 length). If the field depression is (l-p) , the energy limit may be expected to drop by (1-B). At 6=30% the energy limit would be 240 keV. Even 3 a B scaling would only drop the limit to 210 keV. Another deletereous effect arises from the presence of the axial vari­ ation in ambipolar potential. A change in ambipolar potential over many (e.g., 10) gyroperiods along a particle orbit produces only small cor- 3 rections to A|i, while a change in a small fraction of a gyroperiod would produce algebraically, rather than exponentially, small y jumps and hence IV-9 produce low energy limits.
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