I ABB NAVAL RESEARC LAB WAHINGTON DC F/S 18/1.TAB-LI.TY OF.ETA LIITED THERMONUCLEAR BURNIUI IJAN BC E OTT, A MAHIE I UNCLASSIFIED, NRL-MR-4156 SBIE -AD-ElO 38 NL
SECURITY CL AS-SIFICATION o Tsb PAGE "WAo F00e0 OE e REPORT DOCUMENTATIOM PAGE g1po Djc 0 pr g I IMPRp T "16001190 GoVT ACCCSSWN NO: RIEAPaim'S C CATALOG fouuus i NRL Memorandum Report 4156 4. TITL a bis S*ub.e) a. 'V0E OF REPORT S P11091O0 covafeo' STABILITY OF BETA LIMITED THERMONUCLEAR BURN Interim report on a conunuing NRL Problem. s PERVIroH~G Oa4 mgaibtl "MUMOER 2. 7. AUTNO"(0) " CON.TRACT Of GtANT NUMIIUEP48 Edward Ott* and Wallace M. Manheimer 6PERFORMING ORGANIZATION NAME XO ADDRESS IPOGA LM..RJC.Ya" Naval Research Laboratory Washington, DC 20375 NRL Problem 67-08960-0 I I. CONTROLLING OFFICE NAME AND ADDRESS IS REPORT DATE SU.S. Department of Energy January 28, 1980 Washington, DC 20545 Is NuMotR OF PAGES 6 14L MONITORING AGENCY NAME 41 AODRFSS(/i dit.,.ai fer CoGRIVo40#l1 'Olfie) IS SECURITY CLASS (f ONO powi UNCLASSIFIED 1S. 29kASSIVICATOWOOWNGRAIING OULIE Is. DISTRISUTION STATEMENT (o fti Row.foI) Approved for public release; distribution unlimited. * 17. DISTRIUUTION STATEMENT (of the hotfotr t.antord In &leek 20. II dii oreni hot Repoef) IO. SUPPLEMENTARY NOTES *University of Maryland, College Park, Maryland 20742 19. KEY WORDS (Continue oreoe ie, if neooosery.nd identity by block n,,,ibo. Controlled fusion Burn stability 20:\ =@$TRACT fconth", on *,*roe aide It nocobooe and idenify by block.wunboej e burn stability of a thermonuclear reacting plasma Is examined under the assumption that a bum equilibrium exists due to the rapid Increase of loss rate with plasma beta once a critical beta value is exceeded, it is found that perturbations about equilibrium generally result in a rapidly damped exponential decay and a second slow root which can be either growing (unstable) or damped. In the case where the slow root Is growing, the possibility that it can be stabilized by feedback control of the rate at which neutral gas is fed Into the system is considered. DO 147k-501 TIO tof I Nov SSIsl ODSOLrTEa S/N 0102"014"6401 SECURITY CLASSIFICATION OF THIS PAGE (Man Dato ' S ftej L........ - I I! I i I I-i-".... -............... "
STABILITY OF BETA LIMITED THERMDNUCLEAR BURN Plasma heating by fusion products increases with temperature. Thus, once the ignition point is exceeded, in order to establish a thermonuclear burn equilibrium, it is necessary that plasma losses increase sufficiently rapidly with temperature. In this note we examine the implications of the hypothesis that such an equilibrium exists due to a rapid increase in losses with plasma beta once a critical value of beta is exceeded. For example, in tokamaks such a critical beta might result due to the onset of magnetohydrodynamic *instabilities. Since generally accepted theoretical models for anomalous plasma loss in such devices as tokamaks are not available, we shall try to make as few assumptions as possible regarding the dependences of loss rates on plasma parameters except for the existence of a critical beta. We utilize the following zero dimensional model equations for a deuterium-tritium mixture, N-- ync + ONDNT (la) N D y ND+DP NNT N DT (lb) T N + - T D (c) Note: Manuscript submitted November 21, 1979. I,& in.... /i, _II I I................. -..................., - - -
D@ TE N (Tda where is the particle number of species a (a = a, Do Th o for alpha particles, deuterons, tritons, and electrons), y Is the loss rate of particles for species a, is the energy loss rate for species a, O(T) is the deuterium-tritium thermonuclear rate coefficient, Ea is the alpha particle creation energy, E - (N e + ND + NT + NdT is the thermal particle energy content of the plasma, T Is the temperature which has been taken to be equal for all particles (this is approximately true if the equilibration times are sufficiently short compared to the confinement time), and the dot denotes time derivitive. Ohmic heating has been neglected in Eq. (ld) since its contribution is small at the burn point. The quantity S o a denotes the rate at which particles are supplied from an external neutral gas input (the effect of recycling of particles transported to the edge may be absorbed in the coefficient a YN.) Assuming yd = YNT and SD = ST, Eqs. (lb) and (lc) imply that (ND - NT) exp (-tyd d). Thus under these circumstances it is reasonable to take g D -N T and eliminate Eq. (lc). Assuming Na << Ne,ND, neglecting the term ONDNT on the right hand side of (lb), and utilizing the neutrality condition, N. - N + N + 2N, we obtain from Eqs. (1) e D NT+ weotifrmes () N D - y.d + SD (2a) L. T -(e+!)t +eand (2b) 2
--. -. where eo- EI/4 and ye - 1e +.I.D+ Y N" addition, we Y0 2 E 4 R 4 E YN* inadiin shall allow for feedback control of the external Sas puffing rate, SD. We assume that, based upon sensing of the particle number, gas supply rate is adjusted,,d' the D " v [S - F(ND)), (2c) where v -1 is a time constant. This time constant reflects the fact that the control system will have a finite response time, as well as the fact that a particle supplied at the edge will take a finite time to diffuse into the reacting core plasma region. The latter time is probably of the order of and sets a upper limit on v. For simplicity we shall, in what follows, neglect any T or ND dependence of v. From Eqs. (2a) and (2b) the equilibrium burn condition is S D " ' DN RD, (3a) e. 0 ND = (Ye + RDN ) (3b) D e D/D) where the overbar denotes equilibrium. Linearizing Eqs. (2) for -D perturbations about this equilibrium of the form exp (PyNt) we obtain (p + 1 + a1) [p + (Y +1) (1-05) + Ya) (p ++ a 2 6 v -a 2 (p + v) (Y 3 - Y- 2) +a 6 V[ p + (Y+1) (l-a)+ 4 0, 2"' ' ) 4 4) 3 IN :
where y Yn v- y (R / ) ayn/and, a2 -(TI) yi a 3 - (RD/e) aye/and, a4 - (T/'e) 3ye/aT, e - (T/j)dO(T)/dT, and 1 6 = (RD/ D) df(nd)/dnd. In the case where there is no feedback and SD is held constant independent of plasma parameters, Eq.(4) becomes (V40) (p + 1 +a 1 ) (p + (Y+) (-a) + ya 4 ] -a 2 (a 3 - y - 2) - O. (5) We now consider a special case: there is a rapid increase of energy loss (ye) past the critical value of 0, but the particle lose rate D) is relatively uneffected (this situation might result from the onset of magnetic field braiding in a tokamak). In this case a 3 % a 4 >> a 1 %,a 2 (6) (Note that since the a terms are derivative of logarithms with respect to logarithms, much greater than inequalities, such as (6), are in actual situations likely to be marginal at best. Thus the case which we consider should be regarded as illustrative only.) For this situation, Eqs. (5) and (6) show that without feedback there are two roots p - P 1, P 2 ' 1. )p p P1 3- Ya, (7a) 2 2 (a2a3/*4) - (1 )(7b) IN 4 - I I-..... -. - -
The p 1 root represents a rapidly damped solution (-p, " j P 2 ) " The root P 2 can be either damped or $roving depending on the sisn of a and and the relative magnitudes of thea coefficients. We now consider the possibility of stabilizing the root p 2 by feedback (for the case P 2 > 0). Applying (6) and (4) we find that stability results if both -D -D Vp 2 Y D and dfldnd P 2 Y. (8) and the burn is unstable otherwise. Thus we see that stabilization occurs if the system response time is less than the instability growth time, and if, in addition, the feedback i strong enough -D (df/dnd -D Since a lower limit on y results from the finite time for a particle supplied at the edge to reach the reacting core, it is not clear that (8) can be satisfied. Acknowledgement This work was supported by the U. S. Department of Energy. 8 1Mb $Kft t V onmm t.....w - --.. I ll P.5 - - I