J Fusion Energ DOI 10.1007/s10894-010-9332-9

ORIGINAL RESEARCH

Parameter Optimization Studies for a Tandem Mirror Source

W. Horton • X. R. Fu • A. Ivanov • A. Beklemishev

Ó Springer Science+Business Media, LLC 2010

Abstract A basic physics tandem mirror experi- alpha particles born in the core of the plasma with a ment is proposed to develop the potential uses of magnetic relatively large gyroradius. The recent example of the mirror confined plasmas for a neutron source. We consider intrinsically long single particle confinement time is parameter variations from the currently operating sym- given by the 300 s electron confinement time in the metric mirror plasma trap GDT in an attempt to optimize RT1 dipole experiment at the University of Tokyo [1]. the neutron source intensity while minimizing the expense 2. No toroidal curvature that is responsible for the and complications of the system. The combined radial and uniform unfavorable magnetic curvature over large axial plasma loss rates are analyzed and shown to yield an volumes of the plasma. A torus of major radius R has 2 optimal operational point that minimizes the required an effective gravity from the geff = cs/R over the auxiliary heating power. whole outer-half of the torus. This produces a magnetic type of Rayleigh–Taylor instability that Keywords Mirror confinement Tandem mirror limits the plasma pressure to low values in MHD and Neutron source produces long fingers of vortices in the drift wave turbulence that rapidly transport particles and energy radially across the machine’s minor radius r = a. The Introduction favorable and unfavorable magnetic field curvature in the mirror system are next to each other over the

The fundamental understanding of mirror confined plasma distance Lt and, furthermore, vanishes by symmetry in has progressed rapidly in the last 5 years with breakthroughs the core of the plasma. The result is MHD stable in performance from both the Gas Dynamic Trap and the plasma up to a dimensionless plasma pressure of GAMMA-10 systems. There are intrinsic advantages to the b = 8 pp/B2 of order unity. This allows a much higher linear, symmetric mirror confinement geometry over the toroi- plasma pressure to be confined for the standard dal geometry for particle confinement and plasma heating. magnetic field and keeps the volume of the neutron Briefly, some of the advantages of the linear, symmetric source less than 100 m3. confinement system are: 3. The are no large plasma currents as in the system. For steady state tokamak operation the large 1. Infinite confinement times for certain classes of charge toroidal plasma I current must be driven, which particle orbits which may be very important for the p presents an unsolved technology problem. Even when the toroidal current is driven there are spontaneous W. Horton (&) X. R. Fu magnetic reconnection events that can disrupt the core Institute for Fusion Studies, University of Texas at Austin, plasma confinement. Examples, for a burning plasma Austin, TX, USA in a tokamak are given in Horton et al. [2] and Hu e-mail: [email protected] et al. [3]. A. Ivanov A. Beklemishev 4. Mirrors have a natural open magnetic field line Budker Institute of Nuclear Physics, Novosibirsk, Russia divertor that removes the plasma transported to the 123 J Fusion Energ

edge of the cylindrical confinement region. We show a with the required frequency and wavenumbers for the simulation in ‘‘Burning Plasma Pulse’’ where the core resonant wave-particle interactions. plasma is burning and the cooler plasma is transported Molvik et al. [7] propose to design a dynamic trap from the hot core to outer shell where the plasma neutron source (DTNS), which burns approximately 100 g escapes out the machine ends into the expansion tanks. of tritium per full-power year. Such a DTNS would provide The natural scrape-off layer effect is from the higher a neutron spectrum similar to that of ITER and satisfying axial loss rates from pitch-angle scattering in to the the missions of the materials community to test fusion velocity space loss cones in the outer radial zone. materials. 5. The colluminated loss of plasma out the ends of the In the next section, we use a high resolution space-time cylinder then gives the potential of direct conversion of transport code called ITB developed earlier by Horton and the kinetic energy in the plasma to electric power. On Zhu [8] to describe the competition between the radial and the GAMMA-10 device a direct electric convertor has end losses of an axisymmetric mirror system. In ‘‘Confine- been installed by the Osaka University group. The ment Dynamics’’ we give the drift wave turbulence transport converter powers a small neon light bulb from the formulas, and we consider optimization of the system separation of the escaping ions and electrons in cusp parameter vector Pa ¼ ½a; Lcc; Lt; Bcc; Bmax; PICRF; PECRF; magnetic field geometry. Pcc; NBI; hNBI; Pplug;NBI. The parameters start at those for the present GDT and also we discuss how to use genetic algo- Simonen points out [4] that the results from the Gas rithms (GA) adopted from earlier research [9] to find optimal Dynamic Trap in Russia and the GAMMA-10 tandem mirror combinations of system parameters within this multidi- in Japan have renewed interest in concepts. mensional parameter vector space. The equivalent neutron The Russian Gas Dynamic Trap experiment achieved flux of order 1018/s over the targets of 1 m2 corresponding to plasma b = 0.6 in an axisymmetric magnetic mirror. 2 MW/m2 is expected in for these values. In the last section, The Japanese Gamma-10 experiment demonstrated the we give the conclusions. suppression of radial transport due to drift-wave turbulence. Simonen notes that the simple axisymmetric configuration has applications ranging from a source for material testing with plasma and to a driver for fusion-fission The Combined Axial and Radial Transport Code hybrid to the possibility of a plant. for the Mirror System Ivanov et al. [5] review the status of the experiments on the axially symmetric magnetic mirror device gas dynamic With the transport code we varying the ECH heating power trap (GDT). The plasma has been heated by skewed in response to the core electron temperature so as to injection of 20-keV, 3.5-MW, 5-ms deuterium/hydrogen maintain a targeted core electron temperature. The simu- neutral beams at the center of the device, which produces lation varies the core heating power every Dt in response to anisotropic fast ions. Neither enhanced transverse losses of the measured core electron temperature. We propose that the plasma nor anomalies in the fast ion scattering and this real-time coupled diagnostic and auxiliary heating slowing down were observed. Extension of neutral beam system be developed and tested on the University of Texas injection pulse duration from 1 to 5 ms resulted in an Helimak plasma that is created in Argon with microwave increase in the on-axis transverse beta (ratio of the trans- heating and then exported for experiments on G-10 and verse plasma pressure to magnetic field pressure) from Gas Dynamics Trap. 0.4 at the fast ion turning points near the end mirrors to The transport equations for the four fields ne, na and Te, about 0.6. The measured beta value is rather close to or Ti are written as follows: even higher than that expected in different versions of the o 1 o n þ ðrhn v iÞ ¼ S J ð1Þ GDT-based 14-MeV neutron source for fusion materials ot e r or e r ne jje testing. The density of fast ions with the mean energy of o 1 o 19 -3 n þ ðrhn v iÞ ¼ n n hrvi J ð2Þ 10–12 keV reached 5 9 10 m near the turning points. ot a r or a ra D T DT jja The electron temperature at the same time reached 200 eV. The radial plasma losses were suppressed by sheared where Jjjðr; tÞ are the axial fluxes due to the pitch angle plasma rotation in the periphery driven by biasing of end scattering. The pressures pa from Te and Ti evolve wall segments and the radial limiter in the central solenoid. according to Zhmoginov and Fisch [6] show that it may be possible in o 3 1 o 5 a mirror machine to transfer the kinetic energy in the alpha o pa þ o rqa þ TaCa ¼ Paux þ Pa Pajj particles directly to the ions by the proper design of RF t 2 r r 2 antennas to create a finite amplitude ion cyclotron wave ð3Þ

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The sources Sa, Pa and sinks Qa, J||a, Pa|| are specified from with /i and /e measured in electron volts converted to 3 both the auxiliary fueling and heating and the self-nuclear Joules for P|| as power density in MW/m . fusion heating and power transfers between the ions and Two reference models are given in Table 1 for the Proof electrons. The axial losses J|| and P|| are from pitch-angle of Principle (PoP) machine, the DT neutron source(DTNS) scattering for which the confinement time is sjj ¼ machine and the associated scale-up factor for going from pffiffi p Rm the PoP to the DTNS in the fourth column of the table. exp ðciÞðApcisii þ RmL=vth;iÞ and Ap ¼ lnð2Rm þ 2Þ 4 Rmþ1 The two standard drift wave turbulent diffusivities v of ¼ 1:5 where R = B /B is the mirror ratio and A is the m plug p energy are Pastukhov parameter. T 130 V Reference parameters are defined in Table 1 such that vB ¼ cB e ¼ cB ¼ cB464 m2=s ð9Þ ci = /i /Ti = 5.75, and ce = /e/Te = 8.0 as given by Hua B :28 T and Fowler. The central cell electron density is denoted by q T 5:9mm130 V vgB ¼ cgB s e ¼ cgB ¼ cgB2:7m2=s ð10Þ n, temperatures by Te and Ti. The practical formula for the LTe eB 1m :28 T ion-ion collision time is 3/2 We note that the parametric dependences of Te/B and Te / 2 1=2 3=2 3=2 2 66:820mi Ti TiðkeVÞ (aB) for the radial transport explain the energy confine- sii ¼ 1=mi ¼ 4 5:2ms ; ð4Þ nie lnKi n19lnKi=20 ment time scalings in the ITER database. In Fig. 2 we show the radial profiles of a plasma with using Eq. (11.24) of Goldston and Rutherford [10]. For time-constant auxiliary heaters P (r) and P (r) for specified n = 1020 m-3 and T = 10 keV, we have s = 4.5 ms e i i ii radial profiles. R L/v = 0.43 ms. m th,i The transport code evolves the profiles of heating and Formulas for thermonuclear heating power P , radial a cooling. In Fig. 2, we show the results using a real-time power loss Pradial, and end loss power Pjj are: control of the auxiliary heaters Pe (t) and Pi (t) that change 1 2 with a feedback rate an and aTe very Dt * 1 s such that the Pa ¼ nDnT hrviDT Ea ¼ n hrviDT Ea ¼ Pa=e þ Pa=i ð5Þ 4 temperature is returned to the target temperature Te and Ti 3nðT þ T Þv and density n. For the radial fluxes we use drift wave P ¼ e i ð6Þ radial 2a2 formulas. The simplest procedure is to use either the Bohm and gyro-Bohm formulas while recognizing that the n / c n T P ¼ i i ¼ i i i : ð7Þ toroidal confinement database indicates confinement that ijj s s jj jj falls between these two laws in its dependence at the

ne/e ceneTe parameter. Pejj ¼ ¼ : ð8Þ sjje sjje The MHD stability of the axisymmetric mirror reactor is maintained by the large expander cells at each end of the device. The schematic of the expander cells from Pratt and Table 1 Machine parameters Horton [11] is shown here in Fig. 1. This design follows Parameter PoP value Reactor value Scale-up for PoP a 1 m 1.5 m 1.5 L 7 m 30 m 4.28 n1020 m-3 1020 m-3 1

Bcc .28 T 3 T 10.7

Bplug 2.5 T 18 T 7.2

Te 600 eV 10 keV 16

Ti 32.5 eV 15 keV 461 Gas type Deuterium Deuterium and tritium 1.5 Volume 22 m3 212 m3 9.6 Surface area 44 m2 283 m2 6.4 ci = /i/Ti 5.75 7.8 1.35 ce = /e/Te 8 8.5 1.06 3 3 PECH .8 MW/m .8 MW/m 1

Rm 99 1 be .066 .27 4 Fig. 1 A flux surface of the KSTM from Ref. [11] 123 J Fusion Energ that proposed by Post [12] and has now been investigated with 20% from the radial transport and 80% from the axial in detail by Pratt and Berk for interchange stability [13]. transport. The system is stable to the flute interchange mode when sufficiently good connection exists between the end cell A Set of Analytically Derived Parameters plasma and the central cell plasma. This requires sufficient density in the end cell. From these consideration we arrive at the neutron source parameters Confinement Dynamics sE ¼ 0:1s ð16Þ 3 W ¼ nTpa2L 108 J ð17Þ Now we consider the combination of radial transport and 2 e the axial transport, we find that there is an optimal auxiliary 9 Pinj 10 W ð18Þ heating power P and the associated plasma electron 2 2 temperature Te to obtain the highest confinement at the pa B 12 m T ¼ 12 Wb ð19Þ lowest auxiliary injected power for the mirror system. Here L 30 m ð20Þ we derive the values P; Te from analytic analysis keeping 3 3 the other system parameters fixed. To allow multiple cost Vp ¼ 100 m cp700 m ð21Þ functions in addition to injected power, over all system p ¼ 3 105 Pa ð22Þ variables we propose to use the Genetic Algorithm (GA) 7 for optimizing object functions as carried out for space pmag ¼ 8 10 Pa ð23Þ physics model in Spencer et al. [9]. This GA method W 30 MJ ð24Þ requires large computational effect, however. p 12 3 nesE ¼ 10 s/cm ð25Þ Optimization of the System Qfus 0:1to0:5 ð26Þ

We define the ‘‘object function’’ as the injected power that Pfusion=Pinj 10 MW=100 MW ð27Þ equals the loss power in the steady state. To find the lowest injected power we vary the electron temperature which is a key parameter for the neutron source [14]. To find the optimal Burning Plasma Pulse heating power P for fixed other parameters for the tandem When the auxiliary heating power exceeds the critical mirror system we calculate the minimum of Ploss given by P 2 power there is a burning pulse or wave that propagates Ploss ¼ðradial fluxÞð2paLÞþðend lossÞðpa B=BmaxÞ from the core of the plasma. Here we show an example of ð11Þ the burning pulse and the associate axial loss rate. Figure 2 shows the evolution of the particle end loss rate with the radial flux given by as the plasma heats up from below 1–20 kev for the nT qe ¼ðc v þ c v Þ e ð12Þ parameters of the experiment. The vertical solid line at r B B gB gB a where vB = Te/eB and vgB = (qs/a)vB. The simplest result(for cgB = 0is c 2=5 T T k nZKa2B 2=5 13 e ¼ð eÞminPloss ¼ ð Þ ð Þ 4cB for which confinement time is 3=5 2=5 4=5 1=5 3=5 1 ð4cBÞ c c ð4cBÞ ðnZKÞ ¼ k þ k ð14Þ s 2 3=5 2 1=5 E minPloss ð4a BÞ ð4a BÞ and the required injected power at this electron temperature is Fig. 2 The evolution of the particle end loss rate as the plasma heats 1=5 4=5 4=5 up from below 1–20 kev. The snapshots are marked by letters with cB ck ðnZKÞ Pmin=ne ¼ 1:7 ð20% þ 80%Þð15Þ At= 0, Bt= 1s,Ct= 2s,Dt= 3s,Et= 4 s, and Ft= 5s. ð4a2BÞ1=5 The units are based on the parameters of (16)–(27)

123 J Fusion Energ q = r/a = 1 defines the cool edge plasma where atomic operating point would require feed back control to guard recombination would begin. against thermal collapse. The radial transport provides a stable operating point in temperature space once the temperature exceeds a mini- Conclusions mum critical value. The requirements for optimizing the neutron source The combination of axial and radial transport for the rather than energy multiplication as in the ITER project are symmetric mirror system leads to an attractive DT burning different as explained by Ryutov [14]. The mirror system plasma. The global confinement time s is shown to be a has definite advantages over the torus as a compact effi- E hybrid of the core radial transport coefficient and the axial cient neutron source. The neutron source with a flux of 14 2 2 pitch angle scattering time sjj: An optimization is presented fn = 1.4 9 10 n/cm s or 3MW/m can be produced by the injection of 200 kev triton beam into a thermal plasma in Eq. 15 that minimizes the power loss Pmin. The depen- dence on the system parameters is derived for the simple with Te \ 1 kev. The system can be less than L = 10 m and radius less than a = 30 cm according to Ryutov’s Bohm transport formula which shows that 20% of the design [14]. The high efficiency of the system compared power lost is from radial transport and 80% from axial with a torus comes from the theory and experiments losses. showing that a mirror confined plasma is stable up to a relative plasma to magnetic pressure of order unity. In the torus the large effective gravity, as explained earlier, limits References the plasma pressure to less than 10% of the magnetic pressure. 1. Z. Yoshida, H. Saitoh, J. Morikawa, Y. Yano, S. Watanabe, Y. 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