Thermal Stability of Pure and Catalyzed

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 16[9], PP. 619~636 (September 1979). 619 Thermal Stability of Pure and Catalyzed D-D Fusion Reactor Plasmas Yasuyuki NAKAO, Masao OHTA and Hideki NAKASHIMA Department of Nuclear Engineering, Kyushu University* Received April 3, 1978 Revised February 27, The thermal stabilities of the D-D fusion reactor plasmas operating on the puredeuterium and the catalyzed-deuterium cycles have been analyzed by the first-order perturbation method in order to make clear the steady-state operating conditions. When the constant or trapped-ion mode scaling of the confinement-time is used, the critical temperature of the self-heated plasma lies in the vicinity of the minimum point for the confinement parameter nt. Bohm scaling gives the lowest value for the critical temperature, while neo-classical scaling does the highest one. With increasing fuel injection energy, the growth rate of the instability at low temperature decreases, whereas it does not always decreases at high temperature. In the D-D fusion plasma, the subsidiary reaction 3He(d,p)4He makes dominant contribution to the plasma heating and the instability at low temperature. The cyclotron radiation, the dominant power loss mechanism at high temperature, plays a significant role in determining the critical temperatures. Calculations neglecting the cyclotron radiation yield the inadequate, considerably higher values for them. The validity of the above results has been confirmed by the non-linear dynamic simulations. KEYWORDS: D-D fusion reactor, pure-deuterium cycle, catalyzed-deuterium cycle, plasma energy balance, stability, first-order perturbation method, critical temperature, confinement-time scaling, subsidiary reaction, cyclotron radiation, fuel injection energy, non-linear dynamic simulation, instability, plasma heating, low temperature, very high temperature, temperature dependence I. INTRODUCTION The D-D fusion reactor plasmas are characterized by the higher operating temperature and the reduced power density compared with the D-T fusion reactor plasmas. Nevertheless, D-D reactors are considered to be of long range interest for their several potential advantages : almost limitless supply of deuterium fuel, flexibility in the blanket designs, simplified tritium handling and reduced radiation damages. A series of previous studies(1)~(4) discussed the nuclear characteristics of D-D reactor blankets. Furthermore, the study of the reactor plasma characteristics is of course needed for the attainment of the technological feasibility of the reactors. The present paper concerns the stability of the equilibrium D-D fusion reactor plasma against the perturbations in the particle densities and temperatures-thermal stability. The information of this type is essential to make clear the steady-state operating conditions. The problem of such a stability including the dynamic characteristics of the reactor plasma was first discussed by Mills(5), then developed by Ohta et al.(6)~(8), and has * Hakozaki, Higashi-ku, Fukuoka

2 620 J. Nucl. Sci. Technol., received considerable attention(9)~(25). However, the most of them dealt with D-T reactors. A few researchers reported the stability or dynamic characteristics of the D-D reactor plasma operating on the pure-deuterium cycle by making use of the simple plasma models. Powell & Hahnc(22) examined the stable temperature range of D-D reactor plasma by solving the time-dependent particle and energy balance equations under the assumptions of constant confinement-time. Nakashima & Ohta.(23) discussed the effect of the confinement-time scaling on the stability using the similar method. The constant, Bohm and classical types of scaling were considered. Usher & Campbell(24)(25) compared the critical temperatures among the different fuel cycles. All of these calculations assumed that the electron temperature is the same to the ion temperature, and neglected the energy loss of electrons due to the cyclotron radiation. The high operating temperature of the D-D fusion reactor plasma, however, suggests the significance of the cyclotron radiation in determining the upper limit temperature of the possible operating range and the critical temperature. In the present paper, the stability of the D-D fusion plasma is analyzed on the basis of more realistic plasma model. The trapped-ion mode instability is also included in the confinement-time scaling. The perturbation method developed by Ohta et al.(6) is applied with some extentions. The effects of the cyclotron radiation and the subsidiary reactions on the stability are made clear. Two operating modes are considered : puredeuterium (pure D) cycle and catalyzed-deuterium (cat. D) cycle. The concept of cat. D cycle was introduced by Mills(26) in order to increase the power density in the plasma, where tritium and 3He produced by the primary D-D fuel are recycled and burned at the same rates they are produced. So far, the thermal stability analysis has not been performed for cat. D cycle. Pure D operational mode does not recycle tritium and 3He. The steady-state parameters are first determined, and then the stability of equilibrium plasma is analyzed after the linearization of the basic balance equations. The validity of the results is examined by the non-linear dynamic simulations. I. CALCULATIONAL MODEL 1. Assumptions The following assumptions have been adopted in the present (1) Only the following reactions are important: D+D T+p+Q1, Q1= 4.03 MeV D+D 3He+n+Q2, Q2=3.27 MeV DH+T 4He+n+Q3, Q3=17.59 MeV calculations. D+3He 4He+p+Q4, Q4=18.35 MeV (2) Charged fusion products distribute their initial kinetic energies instantaneously among field ions and electrons through Coulomb collisions. Similarly, ions and electrons injected into the plasma equilibrate instantaneously with like particles in the plasma. (3) Fusion-neutrons escape out of the plasma having their initial energies, and do not concern the plasma energy balance. (4) The particle confinement-time is the same for all ionic species. The density of 2

3 Vol. 16, No. 9 (Sep. 1979) 621 electrons is determined by the charge neutrality requirement. (5) The plasma volume and the magnetic field strength are maintained to be constant. (6) The plasma is spatially homogeneous and is described by the point-kinetic model. 2. Basic Equations* The particle balance equations can be written by taking into account the injection rates, the loss or production rates due to fusion reactions and the escaping rates, as follows: (1) (2) (3) (4) (5) where n n2, n3, n4, and n5 are the number densities of protons, deuterons, tritons, 3He and 4He nuclei, tp, the particle confinement-time, the reaction rate parameter <su>, (j=1~4) the product of the fusion cross-section and the relative velocity, averaged over a Maxwellian distribution, for which use is made of Greene's values(27) in the present calculations, and S2, S3 and S4, the number of deuterium, tritium and 3He atoms injected into the plasma per unit volume per unit time. In pure D cycle, S3=S4=0. In cat. D cycle, S3=(n3/tp)0 and S4=(n4/tp)0, where subscript 0 indicates the steady-state. The energy balance equations for ions and electrons can be written as follows: (6) (7) where Einj is the fuel injection energy, charged particle and Ei the initial energy of Ei transfered to the plasma electrons, Ri of and the production rate for the the particle. fi represents is given by(28) i-th the energetic fractions (8) (9) where Zi is the ion charge, T, the electron temperature and A; the ion mass number. ne is the electron density determined by the charge neutrality requirement, * All quantities are in MKS units, except temperatures, which are in kev. 3

4 622 J. Nucl. Sci. Technol., (10) where ni and ZI are the density and atomic number of any impurity present in the plasma. All plasma ions are assumed to be at the same temperature Ti. Sf indicates the sum taken over all charged fusion products, while Sin is taken over all plasma thermal ions. The term Wex represents the energy transfer rate due to Coulomb collisions between plasma thermal ions and electrons, and is given by (11) Wb and Wc represent the energy loss rates of electrons due to bremsstrahlung and cyclo. tron radiation, respectively, and are given by Equation (13) is equivalent to the equation used for the cyclotron loss in Ref. (29), which combined the techniques developed by Rose(28) and Krajcik(30). B is the magnetic field in the plasma, l the characteristics dimension of the plasma, and l=4m is assumed in the present calculation. The dimensionless factor Kc represents the fraction of cyclotron radiation that is ultimately absorbed in the first wall, and is given by where a0, a1 and a2 are given in Ref. (30). b is the ratio of the plasma kinetic pressure, and given by (13) (14) (15) where m(=4px10-7) is the permeability of free space. b provides some measure of the efficiency with which the magnetic field is being utilized(10). 3. Confinement-time Scalings In order to analyze the thermal stability, it is necessary to know the dependence of confinement-times on particle densities and temperatures, which is not known sufficiently at present. In the present study, several different models of confinement-time scaling are considered, a closed magnetic system being assumed : Bohm diffusion, neoclassical diffusion and trapped ion mode instability (TIM). (a) Bohm scaling (16) (b) Neo-classical scaling(31) (17) (c) TIM scaling(32) (18) (19) For the sake of comparison, constant scaling is also considered where confinement-times are independent of both densities and temperatures. 4

5 Vol. 16, No. 9 (Sep. 1979) 623 According to the Neo-classical diffusion theory, the relative magnitude of tp to tei is given by (20) where me is the electron mass and M the average ion mass. For D-D plasma, tp~ Q60tEi, In this case, it is impossible to obtain the steady-state solutions because of the enhanced radiation loss resulting from the excessive accumulation of reaction products(12), especially He and P. It has been found that the energy balance becomes possible when 4 the relative magnitude of pp is reduced totp~<6tei. In the present study, discussions are made under the condition t=tei=ee. This assumption corresponds to the case where thermal conduction is ignored. III. CALCULATIONAL METHOD 1. Steady-state Solutions Time-independent versions of Eqs.(1) (5),(10) and (11) are first solved to determine the equilibrium plasma parameters and possible operating range by the following procedure(10): (1) Fix B and b. (2) Solve Eqs. (1)~(5),(10) and (11) for Ti, Te and n2t by using the fact that n1/n2,,wb/n22 etc. can be written in the functional forms as f(n2t, (3) Determine n2 from Eq. (15), then determine t. 2. Linear Stability Analysis The basic balance equations can be written, after chain differentiations of time derivatives in Eqs. (10) and (11) followed by substitutions from Eqs.(1)~(5), in the generic forms as follows: (21) where xj=nj In order the equilibrium (j=1, 2,,5), x6=-ti and x7=te. to examine the stability against perturbations, expansions are made points, about (22) Substituting the linearized Eq. (22) into Eq. (21) and neglecting non-linear terms indxj, simultaneous differential equations with constant coefficients we obtain (23) According to the theory of linear differential equations(33), the stability of solutions to Eq. (23) can be discriminated by the eigen values of the following matrix A: (24) When all of the eigen values have negative real parts, the perturbation dx, will asymp- 5

6 624 J. Nucl. Sci. Technol., totically decay in time : the equilibrium plasma is stable. Conversely, if any eigen value has a positive real part, perturbations will grow in time : the equilibrium plasma is generally unstable. IV. RESULTS AND DISCUSSIONS 1. Steady-state Operating Conditions Equilibrium plasma parameters such as particle densities, electron temperature, required confinement-time and fueling rates have been determined as a function of ion temperature by solving the steady-state balance equations. Two combinations of b and B have been considered : moderate value of b=0.05 with a magnetic field of 10Wb/m2 to represent a conservative condition, and high value of b=0.1 with the magnetic field of 20 Wb/m2 to represent a optimistic condition(10). For purposes of comparison with D-T plasma, the same injection energies as those of Stacey(10) have been considered, that is, 0, 50 and 100 kev. In this energy range, suprathermal reactions have little effect. In practice, however, optimal energy must be determined by taking into account the penetration of injected fuel into the plassma. As an impurity condition, oxygen, which comes from the wall, is assumed to be present in the plasma with density of 0.5% that of the deuterium ion*. (1) Figure 1 (a),(b) shows the confinement parameter n2t as a function of ion temperature. The ignition temperature, at which the curves begin, is about 40keV for pure D cycle, and about 35keV for cat. D cycle. The shape of the curve is primarily determined by the power balance between fusion power carried by charged particles and radiation power : bremsstrahlung at low temperature and cyclotron radiation at high temperature. With the increase of the injection energy, n2t reduces since the (a) B=10Wb/m2, =0.05 (b) B=20 Wb/m2, b =0.1 Fig. 1 Confinement parameter n2t as function of ion temperature Ti * This later. assumption is somewhat simplified. The effect of changing impurity condition is examined 6

7 Vol. 16, No. 9 (Sep. 1979) 625 injected particles having an initial energy need not be confined so long to attain the temperature required for the energy balance(29). In cat. D cycle, when Einj=100 kev, n2t shows the opposite behavior to other cases at low temperature. A similar feature is noted in Ref. (29), where a local minimum of n2t is found at Ti~_2/3Einj. This is because with sufficiently large energy of injection, the injected ions need not be confined so long to attain the required temperature, indicating the decrease of t with decreasing Ti as shown in Fig. 2. The minimum n2t in cat. D cycle are ~1/3 of those in pure D cycle. The 1/2.5 effect of the choice of b and B combinations enters through the calculation of Wc,. n2t increases with decreasing b and B at high temperature, since Wc,db-3/2B-1/2 and increases with Ti. Cat. D cycle (B=10 Wb/m2, =0.05) Fig. 2 Deuterium density n2 and required confinement-time r as function of ion temperature Tt (2) Subsidiary reactions are quite important in obtaining the energy balance of the steady-state of D-D fusion plasma. Table 1 compares the contribution of each fusion reaction to the plasma heating at several temperatures for self-heated cases with B= 10 Wb/m2 and b=0.05. In pure D cycle, at Ti=80keV near which the minimum n2t point lies, about 80% of tritium produced are burned by T(d, n)4he reaction, contributing about 18% of the total fusion power deposition in the plasma. Although only about 35% of 3He produced are burned by 3He(d, p)4he reaction, this reaction contributes more than 50% of the total plasma heating. In cat. D cycle, all of tritium and 3He produced are burned up. At Ti=~80keV, T(d, n)4he and lie(d, p)4he reactions contribute 11 and 72% to the total plasma heating, respectively. Table 1 Contribution of each fusion reaction to total fusion ower deposition in (B=10 Wb/m2, b=0.05, Einj=0keV) In spite of the small reaction rate, the subsidiary reaction 3He(d, P)4He is very important for the plasma heating, TH1S is due to the largest Q-value and to the fact 7

8 626 J. Nucl. Sci. Technol., that both of the reaction products are charged particles. The contribution of another subsidiary reaction T(d, n)4he is not so large, since neutron produced carries away about 80% of the fusion power in the reaction. This reaction has a important meaning for the energy recovery in the surrounding blanket. (3) Cyclotron radiation plays a significant role in the energy balance in high temperature region. Figure 3 compares the output power fractions among radiations and escaping charged particles. In this figure, fpi(fpe) is the power fraction carried by escaping ions (electrons), and given by the following equations (25) (27) (27) (28) (29) fpb and fpc represent the power fractions carried by bremsstrahlung and cyclotron radiation, respectively, and given by (30) (31) Cat. D cycle (B=10 Wb/m2, =0.05, Einj=0keV) b Fig. 3 Output power fractions among radiations and escaping charged particles as function of ion temperature Ti With increasing ion temperature, fpb decreases since the decreasing ion densities overshadows the effect of increasing electron temperature. fpi and fpe have broad maximums at Ti=70~80keV. fpc shows a rapid increase with increasing ion temperature because of its strong dependence on electron temperature. Cyclotron radiation is the dominant power loss mechanism at high temperature, and then becomes a key factor to determine the upper limit temperature of the possible operating range, although it has little effect on the ignition temperature or n2t in the lower temperature region. For example, if cyclotron radiation is neglected in the calculations for the self-heated cases, the upper limit temperature of the possible operating range is extended from 110~140keV to about 290keV in pure D cycle, and from 155keV to about 340keV in cat. D cycle. 2. Stabilities of Equilibrium Fusion Plasmas Eigen values of the matrix A defined by Eq. (24) have been calculated as a function of ion temperature. Figures 4 (a),(b) and 5 (a),(b) show the largest real part of them, the growth rate of the instability, for self-heated pure D and cat. D cycles. Negative values indicate the stable equilibrium states, or perturbations which decay in time. Table 2(a),(b) summarizes the critical temperatures which draw distinctions between stable and unstable regions. The attention is first paid to self-heated plasmas. The effect of injection energy on the stability is examined later. (1) The equilibrium plasma is generally unstable at low temperature. The magnitude of the instability decreases with increasing ion temperature. Cat. D cycle gives the 8

9 Vol. 16, No. 9 (Sep. 1979) 627 (a) B=10 Wb/m2, b=0.05, Einj=0keV (b) B=20 Wb/m2, b=0.1,einj=0kev Fig. 4 Largest real part of eigen values Re(E.V.)max, as function of ion temperature Ti- Pure D cycle Table 2 Critical temperatures for pure D and cat. D cycles (kev) (a) Pure D cycle (b) Cat. D cycle * Stable in all of the temperature range considered, ** Stable when Ti<_61 kev or Ti> 77keV lower critical temperature compared with pure D cycle for each of the confinement-time scalings by the following amount : constant 3.10keV, Bohm 13~15keV, neo-classical kev and TIM 7~14keV. Although the growth rate of the instability near the ignition point is larger in cat. D cycle than in pure D cycle, the wider stable operating range in addition to the reduced confinement requirement obtained for cat. D cycle may support the superiority of cat. D cycle to pure D cycle. When the constant or TIM scaling is used, the critical temperature lies in the 9

10 628 J. Nucl. Sci. Technol., (a) B=10Wb/m2, b=0.5, Einj=0keV (b) B=20Wb/m2, b=0.1, Einj=0keV Fig. 5 Largest real part of eigen values Re(E.V.)max, as function of ion temperature Ti Cat. D cycle vicinity of the minimum n2t point*. Bohm scaling gives the lowest value for the critical temperature, while neo-classical scaling does the highest one. A comparison between Bohm and neo-classical scalings can be made by the following simple example. Consider positive perturbations in fuel ion densities at unstable, low temperature point. By increasing the fusion reaction rates, these increase ion and electron temperatures. In Bohm scaling the increased electron temperature, which decreases confinement-time, tends to suppress the inital perturbations. In fact,aj7<0 (j=2, 3, 4) negative feedback is obtained. Neo-classical scaling, on the other hand, acts to enhance the initial perturbations since the increased electron temperature increases the confinement-times : aj7>0 (j=2, 3, 4)- positive feedback- is obtained. TIM scaling is the least unstable near the ignition point. However, the growth rate of the instability decreases more slowly than those in the other scalings with in - creasing ion temperature. As a consequence, TIM scaling gives relatively high values for the critical temperature. It should be noted that in D-T reactor plasma TIM scaling gives the lowest critical temperature (12). This discrepancy could be understood by the fact that the critical temperature of D-T fusion plasma lies below the ignition tem - perature of D-D fusion plasma. (2) The principal mechanism responsible for the instability at low temperature is the * The minimum of n2t occurs at Ti=85~95keV in pure D cycle, and at Ti=75~85 kev in cat. D cycle. 10

11 Vol. 16, No. 9 (Sep. 1979) 629 rapid variation of the reaction rate parameter with ion temperature(10) : at low temperature, a change in temperature causes the greater change in the fusion power than in the rate of energy loss by radiations and by escaping charged particles(22), which makes the fusion plasma unstable. This could be said independently of the type of fuel cycle. The effect of each fusion reaction on the instability can be compared by the calculations setting (d/dti)<su>i=0 (j=1, 2, 3 or 4). Table 3 summarizes the results for the self-heated pure D cycle. Compared with the actual case where (d/dti)<su>j=/0, the calculations setting (d/dti)<su>j=0 Table 3 Critical temperatures obtained by calculations setting (d/dti<su>j =0 (j=1, 2, 3 and 4) (kev) (1=1, 2 or 4) reduces the growth rate of the instability, resulting in the lower values for the critical temperature. Especially in case of (d/dti)<su>4=0, the critical temperature is lowered for each of the confinement-time scalings by the following amount : constant 26% ; neo-classical 12% and TIM 30%. Bohm scaling is stabilized in all of the temperature range considered. The calculation setting (d/dti)<su>3=0 yields the slightly higher values for the critical temperatures, although it decreases the growth rate of the instability at low temperature where Ti<~70keV. Pure D cycle(b= 10 Wb/m2, b=0.05, Einj=0keV) ss Case (d/dti)<su>i=0 Stable in all of Constant Scaling Bohm Neoclassical ( j=1, 2, 3 and 4) the temperature range considered Within the operating temperature range determined in the present calculation D(d,p)T, <su>2 and <su>4 increase with increasing ion temperature. Then, the reactions D(d, p)t, D(d, n)3he and 3He(d, p)4he act to make the fusion plasma unstable in all temperature range considered. The subsidiary reaction 3He(d, p)4he has the largest effect on the instability, since the contribution to the change in the total fusion power deposition in the plasma is the largest from this reaction. The <su>3, on the other hand, increases with increasing ion temperature up to about 70keV ; for Ti>~70keV, <su>3 decreases with ion temperature. Then, another subsidiary reaction T(d, n)'he contributes to make the fusion plasma unstable at low temperature, and for Ti>,~70keV, in turn, tends to make the plasma stable : the calculation setting (d/dti)<su>3=0 slightly increases the critical temperatures. The magnitude of this effect, however, is not significant since the variation of <su>, is not so rapid within the temperature range of interest and the reaction makes the smallest contribution to the total fusion power deposition in the plasma. (3) The D-D fusion reactor plasma is characterized by its high operating temperature. The cyclotron radiation strongly depends on the electron temperature. This fact suggests the significance of cyclotron 11 TIM Table 4 Critical temperature of self-heated fusion plasma obtained by calculations neglecting cyclotron radiation (kev) Calculation model Present model Constant Pure D cycle 132 Cat. D cycle 98 Powell et al.(22) ~ 100 (pure D cycle) Nakashima et al.(23) 104 (pure D cycle) Campbell et al.(21) 82 (pure D cycle) Scaling Bohm Neoclassical > TIM

12 630 J. Nucl. Sci. Technol., radiation in determining the critical temperature of the D-D reactor plasma. Table 4 represents the critical temperatures obtained for self-heated pure D and cat. D cycles from the calculations setting Wc=0. The effect of the combination of B and b does not appear. Table 4 also contains the results of previous studies with simple plasma models for pure D cycle. In the previous calculations Te=Ti, was assumed as well as Wc.=0. Considerable differences observed between the present and the previous results would come from the difference in the calculational models adopted. The calculations neglecting the cyclotron radiation increases the growth rate at high temperature, compared with those including the cyclotron loss, resulting in considerable higher values for the critical temperature. Especially in case of pure D cycle, the present model setting Wc=0 increases the critical temperature for each of the confinement-time scalings by the following amount : constant 35~47keV, Bohm 4 kev, neoclassical 155~474keV and TIM 34~47keV. The strong dependence of cyclotron radiation on the electron temperature acts to stabilize the equilibrium plasma at high temperature. For instance, consider the positive perturbation in ion temperature. This increases the fusion reaction rates and then the electron temperature. At high temperature, the increased radiation loss overcomes the increased fusion power, diminishing the initial perturbations. Some previous studies(10)(12)(18)~(21) on D-T plasma assumed the fraction of cyclotron radiation reflected by the metallic wall to be constant. In the present study the absorption coefficient Kc, the fraction of cyclotron radiation absorbed in the wall, has been calculated in a consistent way on the basis of Krajcik's method. K which is a function of Te and ne, takes the value of ~10-3 within the rage of D-D reactor plasma. In order to examine the effect of the absorption of cyclotron radiation in the wall, the thermal stability analysis have been performed with Kc changed by factors of 2, 5 and 10. Figure 6 shows the results of calculations. As expected, with the increase of the magnitude of Kc, plasma becomes more stable, while confinement requirement becomes severer. The calculation with Kc increased by a factor of 2 decreases the critical temperature for each of confinement-time scalings by the following amount : constant 6keV, Bohm 1keV, neo-classical 8keV and TIM 4keV. (In contrast, Gralnick & Tenney(34) reported that in D-T reactor plasma the increase of cyclotron radiation loss has little effect on the determination of the critical temperature since at the lower temperature regime required for optimal operation the plasma is not a strong cyclotron radiator: transport and bremsstrahlung losses dominate.) Pure D cycle (B=10 Wb/m2, b=0.05, Einj =0keV) Fig. 6 Effect of increasing absorption coefficient Kc on thermal stability Summarizing, cyclotron radiation is a key factor to stabilize the fusion plasma o peraung at hign temperature, ana tnen has signincant errects on tne critical temperature 12

13 Vol. 16, No. 9 (Sep. 1979) 631 of D-D fusion reactor plasma. The calculation neglecting it will yield inadequate, considerably higher values for them. (4) The effect of the fuel injection energy on the critical temperature is shown in Table 2 (a),(b). Figure 7 (a),(b) shows the largest real parts of the eigen values in case of Einj=100keV. The growth rate of the instability at low temperature decreases with increasing injection energy, resulting in the lower values for the critical temperature in constant and Bohm scalings. In case of pure D cycle, the growth rate at relatively high temperature does not always decreases, and rather increases, giving slightly higher critical temperatures for neo-classical scaling (cf. Fig. 7(a) us Fig. 4(a)). B=10Wb/m2, b=0.05, Einj =100keV Fig. 7 Largest real part of eigen values Re(E.V.)max, as function of ion temperature Ti When the injection energy of 100keV is used for cat. D cycle, the equilibrium plasma governed by constant, Bohm or TIM scaling is stabilized in all of the temperature range considered. When neo-classical scaling is used, the instability appears only in the intermediate temperature range (~60keV<~Ti<77keV)*. The equilibrium point in the vicinity of the ignition temperature, which is the most unstable in the self-heated cases, is stabilized. This can be attributed to the greatly reduced confinement-time at low temperature (See Fig. 2). For instance, consider changes in temperatures. The induced change in the rate of the energy loss by escaping charged particles exceeds that in the fusion power, by which the plasma is stabilized. * The instability disappears when >~110keV. 13

14 632 J. Nucl. Sci. Technol., In the above discussions, oxygen which comes from the wall, has been assumed to be present in the plasma with density of 0.5% that of deuterium ion. In practice, however, the other impurity would also enter the plasma as a result of sputtering. In order to examine the effects of other impurity species on the thermal stability, analysis has been performed by changing the impurity atomic number Z1 from 8 (oxygen) to 6(carbon), 14(silicon), 26(iron) and 41(niobium). Carbon and silicon would be present in a device with a graphite or silicon-carbide first-wall surface. Iron would be found in a device with a stainless-steel first-wall. Niobium is a representative of high-z materials that might be used for the first-wall. Figure 8 shows the results of calculations. The main effect of impurity is to enhance the radiation power loss. As anticipated, the value of nt required for ignition increases with increasing Z1. For Z1 greater than 15, equilibrium solution can not be obtained unless the relative density of impurity f1 is reduced below 0.5%. The plasma generally becomes more unstable with increasing Z1. In neo-classical scaling, however, enhanced radiation loss at higher temperature (Ti>~90keV) decreases the critical temperature. At any rate, changing the impurity atomic number produces little effect on the critical temperature except for Bohm scaling within the range considered here. When carbon is considered instead of oxygen as an impurity, the critical temperature decreases for each of confinement-time scalings by the following amount : constant 1keV, Bohm 5keV, neo-classical 0keV and TIM 0keV. following Pure D cycle (B=10 Wb/m2,b=0.05, Einj=0keV) Fig. 8 Effect of changing impurity condition on thermal stability Stacey performed the linear stability analysis for the D-T reactor plasma under the conditions(12): Case 1: Only density variations are considered. Linear stability analysis is performed on the particle balance equations setting dti=dte=0. Case 2: Only temperature variations are considered. Linear stability analysis is performed on the energy balance equations setting dn,=0(j=1,2). Case 3: Both of density and temperature variations are considered. Linear stability analysis is performed on the particle and energy balance equations. Comparing the eigen values obtained in each case, he identified the unstable modes at low temperature as pure temperature fluctuations for constant and neo-classical scalings, and as a mixed temperature-density fluctuations for TIM scaling. Similar methods have been applied to the D-D fusion plasma in the present study. However, it has been impossible to make such a identification for any scaling of the confinement-time. This might indicate the strong interactions between temperatures and densities. 14

15 Vol. 16, No. 9 (Sep. 1979) 633 Although the linear stability analysis can make predictions about the instability or stability of the system against perturbations, for D-D fusion plasma it can not say which component n1, n2,.., Ti or Te) tends to become unstable or remain to be stable. Such a identification can be made by solving the non-linear basic equations. 3. Non-linear Dynamic Simulations To confirm the validity of the linear stability analysis, the non-linear equations (21) have been solved numerically by the fourth order Runge-Kutta method. Several equilibrium points have been considered. The following types of the initial perturbations have been introduced to disturb the equilibrium condition: (1) Perturbations of ±5% in densities (2) Perturbations of ±5% in temperatures (3) Perturbations of ±5% in densities and temperatures. It has been confirmed that the results of linear stability analysis are generally consistent with those of non-linear simulations. TIM scaling, however, seems to be slightly more stable in non-linear simulations than would be expected from linear stability analysis. Figures 9 (a),(b) and 10(a),(b) show the temporal behaviors of ion temperature and total ion densities after the perturbations of ±5% in equilibrium ion densities for selfheated pure D and cat. D cycles. Neo-classical (and constant) scaling shows the monotonic behaviors when the perturbations are introduced. Below critical temperature, the positive perturbation brings the plasma to new equilibrium state, while the negative Pure D cycle (B=10Wb/m2, b=0.05, Einj=0keV) Fig. 9 Temporal behaviors of total ion densities Snj and ion temperature Ti after ±5% perturbations in equilibrium ion densities njo (1= 15

16 634 J. Nucl. Sci. Technol., one extinguishs the thermonuclear reactions. Particle densities are somewhat more stable than temperatures. Fig. 10 Temporal behaviors of total ion densities Einj and ion temperature Ti after ±5% perturbations in equilibrium ion densities njo (j=1~5) TIM (and Bohm) scaling shows the oscillatory time dependences when the equilibrium state is disturbed. At low temperature, the positive perturbation as well as negative one extinguishs the thermonuclear reactions. In TIM scaling, the significant interactions are observed between density and temperature components as Stacey concluded for D-T reactor plasma from the linear stability analysis(12). Compared with D-T fusion plasma, D-D fusion plasma shows slower responses to the perturbations : longer times are required to return to initial states or to evolute to new equilibrium states. This is because the change of the fusion power is not so rapid as in D-T reactor plasma. Summarizing the above results, Table 5 is obtained. The table also includes the comparison with the D-T reactor plasma. V. CONCLUSION The thermal stabilities of D-D fusion reactor plasmas operating on the pure-d and cat. D cycles have been analyzed by the first-order perturbation method. The results 16

17 Vol. 16, No. 9 (Sep. 1979) 635 Table 5 Summary of results of thermal stability analysis for pure D, cat. D and D-T plasmas (Einj=0key) have been confirmed by non-linear dynamic simulations. When the confinement-times are governed by constant or trapped-ion mode scaling, the critical temperature lies in the vicinity of the minimum nt point. Bohm scaling gives the lowest value for the critical temperature, while neo-classical scaling does the highest one. In trapped-ion mode scaling, at low temperature the positive perturbation as well as negative one extinguishs the thermonuclear reactions. The growth rate of the instability at low temperature decreases with increasing fuel injection energy. However, it does not always decreases at high temperature, especially in pure-d cycle. Although the equilibrium point near the ignition temperature are more unstable in cat. D cycle than in pure-d cycle, the wider stable operating range in addition to the reduced confinement requirement obtained for cat. D cycle may support its superiority to pure-d cycle. Detailed calculations have made clear the significances of the subsidiary reaction and cyclotron radiation in D-D fusion reactor plasma. The reaction 3He(d, p)4he makes dominant contributions to the total fusion power deposition in the plasma and to the instability at low temperature. Cyclotron radiation is the dominant power loss mechanism at high temperature. Its strong dependence on the temperature stabilizes the equilibrium plasma at high temperature. Calculations neglecting it yield considerably higher critical temperatures. (1) (2) (3) (4) (5) REFERENCES NAKASHIMA, H., et al.: J. Nucl. Sci. Technol., l4[2], 75 (1977). NAKASHIMA, H., et al.: ibid., 14[12], 916 (1977). NAKAO, Y., et al.: ibid., 15[1], 76 (1978). NAKASHIMA, H., et al.: ibid., 15(7), 490 (1978). MILLS, R.G.: Proc. of the Symposium on Engineering Problems of Fusion Research, LA-4250, (1969). OHTA, M., et al.: "Plasma Physics and Controlled Nuclear Fusion Research" (Proc. 4th Int. (6) Conf. Madison, 1971), Vol. 3, 423 (1971), IAEA. (7) YAMATO, H., et al.: Nucl. Fusion, 12, 604 (1972). (8) OHTA, M., et al.: J. Nucl. Sci. Technol., 10[6], 353 (1973) (9) OHNISHI, M., et al.: Nucl. Fusion, 13, 761 (1977). (10) STACEY, W.M., Jr.: ibid., 13, 843 (1973). 17

18 636 J. Nucl. Sci. Technol., (11) FUJISAWA, T.: ibid., 14, 173 (1974). (12) STACEY, W.M., Jr.: ibid., 15, 63 (1975). (13) ITo, Y., FUJI-IE, Y.: Preprint Annu. Meeting At. Energy Soc. Japan, (in Japanese), A5, (1976). (14) OHNISHI, M., et al.: Proc. 2nd Topical Meeting Technol. of Controlled Nucl. Fusion, Washington (1976). (15) TSUJI, H., et al.: Nucl. Fusion, 16, 287 (1976). (16) SAITO, H., et al.: ibid., 17, 919 (1977). (17) TONE, T.: JAERI-M 7300, (in Japanese), 102 (1977). (18) ETZWEILER, J.F., et al.: ORNL-TM-4083, (1973). (19) McNALLY, F.R., Jr., et al.: ORNL-TM-4617, (1974). (20) McNALLY, F.R., Jr.: ORNL-TM-4647, (1974). (21) HIRAOKA, T., et al.: ORNL-TM-4843, (1975). (22) POWELL, C., HAHN, O.J. : Nucl. Fusion, 12, 667 (1972). (23) NAKASHIMA, H., OHTA, M.: Tech. Rep. Kyushu Univ., (in Japanese), 47, 1 (1975). 24) USHER, J.L., CAMPBELL, H.G.: Proc. 1st Topical Meeting Technol. of Controlled ( Nucl. Fusion, San Diego, California, (1974). (25) CAMPBELL, H.G., USHER, J.L.: Trans. Amer. Nucl. Soc., 21, 45 (1975). (26) MILLS, R.G.: PPPL-TM-259, (1971). 27) GREENE, S.L.: UCRL-70522, (1967). ( (28) ROSE, D.J.: ORNL-TM-2204, (1968). (29) CHU, T., MILEY, G.H.: Technology of controlled thermonuclear fusion experiments and the engineering aspects of fusion reactors, CONF , p.322 (1972). (30) KRAJCIK, R.A.: Nucl. Fusion, 13, 7 (1973). (31) ARTSIMOVICH, L.A.: ibid., 12, 215 (1972). (32) DEAN, S.O., et al.: WASH-1259, (1974). (33) For example, HALANAN, A.: "Differential Equations", (1966), Academic Press.( 34) GRALNICK, S.L., TENNEY, F.H.: Proc. 1st Topical Meeting Technol. of Controlled Nucl. Fusion, San Diego, California (1974). 18