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Experimental Study of Equilibrium in a Bumpy Torus
S. Hiroe J. A. Cobble R. J. Colchin G. L. Chen K. A. Connor J. R. Goyer L. Solensten
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DISTRIBUTION OF THIS DOCUMENT IS UNUIftTTEB ORNL/TM—9520 DE86 014415 Pist. cH^W^O f,g
Fusion Energy Division
Experimental Study of Equilibrium in a Bumpy Torus
S. Hiroe J. A. Cobble R. J. Colchin G. L. Chen Fusion Energy Division
K. A. Connor, J. R. Goyer, L. Solensten Rensselaer Polytechnic Institute Troy, New York
Date of Issue: June 1986
Prepared by the OAK RIDGE NATIONAL I-ABORAI OKY Oak Ridge, Tennessee 37831 operated by OlSTRlBUT,0«0FTHlSD0CU^T^^ CONTENTS ABSTRACT v I. INTRODUCTION 1 n. EXPERIMENTAL RESULTS 3 m. DISCUSSION 15 A. Formation of closed potential contours 15 B. Inward displacement of potential contours 22 C. Electrostatic beta limit 26 D. Force balance 28 E. Explanation of potential deformation 33 TV. CONCLUSION 37 ACKNOWLEDGMENTS 39 REFERENCES 41 iii ABSTRACT Plasma equilibrium in the ELMO Bumpy Torus (EBT)1 was studied experimentally by measurements of the electrostatic potential structure. Before an electron tail population is formed, the electric field is found, roughly speaking, to be in the vertical direction. The appearance of a high-energy electron tail signals the formation of a negative potential well, and the potential contours start to nest. The potential contours are shifted inward with respect to the center of the conducting wall. The electric field between the plasma and the conducting wall forces the plasma inward, balancing the outward expansion force. This force balance provides a horizontal electric field that cancels the concentric radial electric field locally at the separatrix of the potential contour and leads to convective energy loss. v I. INTRODUCTION There has been much interest in the kind of equilibrium that can exist in a closed magnetic-field-line system without a rotational transform. Although there has not yet been an experimental discussion of the equilibrium in an electron cyclotron heating (ECH) bumpy torus, there have been several theoretical discussions.-'3,4 Budker2 has pointed out that the E x B poloidal rotation due to the centripetal electric field is equivalent to the existence of a rotational transform with V(B/qR replaced by E, where vj is the velocity parallel to the toroidal magnetic field, q is the safety factor, and R is the major radius. The equipotential contours4 shift inward from the magnetic axis in the case of the potential well. Hereafter, we will only be concerned with the plasmas whose pressure profiles are hill-shaped and whose potential profiles are well-shaped (with exceptions as noted). Bulyginskii et al.5 qualitatively analyzed a mechanism for the formation of an equilibrium plasma configuration in a current-free toroidal plasma. Sufficiently strong centripetal electric fields for an equilibrium were produced by the charge separation caused by toroidal drift and/or the preferential loss of particles. They also carried out a qualitative analysis of the evolution of a plasma left to itself and tending toward equilibrium in the drift approximation. Closed drift trajectories are at least necessary for the potential contours to nest. Thus, when the potential contours evolve from a lack of equilibrium to closed equilibrium configuration, additional drifts are needed to cancel the outward E x B drift due to a lack of equilibrium. The inertial drift is one of these additional drifts. Bulyginskii et al. also pointed out the relation of the inward shift of the potential center to a conducting vacuum vessel and discussed the electrostatic force balance. 1 2 Popkov6 examined the equilibrium and stability of a charged plasma in a toroidal magnetic field and in an external electric field. He found that only a low-density plasma (ci- 1 Recently, El-Nadi7 analyzed the high-density plasma equilibrium (eL- ls> 1) in a bumpy torus (not a simple torus). The density was assumed to be an arbitrary function of + Te /e In U = $ df/B rather than the potential Thus, many physicists have suggested electrostatic equilibria in closed magnetic field-line systems. The important points of these works are: 1. The plasma equilibrium may be attained with concentric, closed potential contours without a rotational transform (electrostatic equilibrium). 2. A horizontal external electric field is necessary to sustain the equilibrium. 3 3. The equilibrium potential contours are shifted inward with respect to the center of the boundary (metal wall, hot electron ring, etc.). 2 4. The electrostatic beta value of pE = 2 /EE is found to be limited when the potential separatrix forms in plasmas. Besides electrostatic equilibrium, the possible existence of an equilibrium without the electric field in the ELMO Bumpy Torus (EBT) was examined. FreidbergS calculated the analytical toroidal equilibria of the EBT configuration by means of an asymptotic expansion in the amplitude of the "bumpiness." Equilibria containing a magnetic well were found. It was shown that a toroidal plasma shifts inward to counteract the 1/R outward toroidal expansion force. The purpose of this paper is to study experimentally the equilibrium in a particular closed field-line device - the EBT. Experimental results are presented in Sec. II and are discussed in Sec. HI. Concluding remarks are given in Sec. IV. II. EXPERIMENTAL RESULTS A prolonged effort has been made by a group from the Rensselaer Polytechnic Institute (RPI) to obtain electrostatic potentials in EBT by means of a heavy-ion-beam probe.9 Recently this probe has been made capable of measuring two-dimensional equipotential (2-D The EBT1 consists of 24 simple mirrors that are connected toroidally. The major radius is 1.5 m, and the minor radius is ~0.1 m at the mirror throat. The mirror ratio at the magnetic axis is 1.9. Vacuum mod-B contours and magnetic field lines are plotted in Fig. 1. The fundamental and second ORNL-DWG 84-3737 FED MOVABLE LIMITER ALUMINUM WALL> CsCHARGE THOMSON EXCHANGE /SCATTERING SPECTROMETRY 5ch DIAGNOSTIC HARD NEUTRAL X-RAY BEAM MACHINE FUNDAMENTAL RESONANCE SOFT CENTER X-RAY GAS FEED LIMITER RGA BEAM OPTICAL VIEWPORT INTERFEROMETER (4 mm) Fig. 1. Location of diagnostics used for this experiment are schematically indicated in the left half of the figure. In the right half, the fundamental electron cyclotron resonance surface, the second cyclotron resonance surface, and the. magnetic field lines are plotted. Position X = 0 corresponds to the machine center, "a" to the center of the bumpy field, "b" to the "operational position" of the movable limiter, and "c" to the position 1.5 cm inward from the field line intersecting the second cyclotron resonance. 5 electron cyclotron resonance positions in this plot correspond to normal EBT-S operation. The magnetic field strength at the fundamental resonance is 1 T. The maximum magnetic field (B()) in the midplane is 0.725 T, and the maximum magnetic field point (magnetic axis, center of the bumpy field) is located toward the inside of the torus. The inward displacement of this point from the machine center is about 3 cm in the midplane. The center of the mod-|B| contours for the second electron cyclotron resonance in the midplane (ring center) is displaced inward by Ar = 1.5 cm from the machine center (geometrical center of a cavity). To study the boundary conditions experimentally, a movable limiter was installed in one of the 24 cavities. The positions denoted by b and c in Fig. 1 are called the operational position and the inside position, respectively. The heavy-ion-beam probe cavity is located three cavities away from the limiter cavity. A Thomson scattering system was employed to measure the bulk electron density and temperature. It was located 11 cavities away from the limiter cavity. The EBT plasma is produced by high-power microwaves (P^). The frequencies of the microwaves (fM) are 18 and 28 GHz. The device is called EBT-1 when f„ = 18 GHz (B0 = 0.5 T); it is called EBT-S when fM = 28 GHz (B0 = 0.725 T). Typical operation was carried out by keeping constant and changing ambient pressure (p0). When the machine was operated in this manner, there were three different operational modes,1 called the C-mode (high p0), the T-mode, and the M-mode (low p()). The transitions between the individual modes are called the T-C transition and the T-M transition. Stability at the T-M transition has already been discussed.10 The hot electron instability prevents operation at lower pressure. Equilibrium at the T-C transition will be discussed later. 6 Typical midplane 2Dcj> contours for EBT-1 are shown in Fig. 2, as measured for the different operational modes (where the potential contours are labeled in volts). The horizontal direction is denoted by X and the vertical position by Z. The point X = Z = 0 is at the machine center. The torus axis is located at the left side of each graph. The toroidal magnetic field is directed downward into the plane of the paper. The equipotential contours in the C-mode are represented in Fig. 2a, a low-pressure C-mode in Fig. 2b, the T-C transition in Fig. 2c, and a typical T-mode in Fig. 2d. The accuracy in measuring the position near the center is better than 1 cm but is less near the hot electron rings. The absolute accuracy of the potential measurement is ±20 V near the center. During the processing of the raw data, the potential value was taken to be zero at X = ±15 cm and Z = ± 15 cm. We do not discuss the potential contours outside the region that are shown by the broken line in Fig. 2a because the accuracy is poor. Note that the location of this circle is close to the location of the second cyclotron harmonic resonance shown in Fig. 2d. Typical potential structures are summarized as follows. 1. The electric field in the C-mode consists of a vertical part (Ev) and a horizontal part (Eh). In EBT-1 (Fig. 2a), Ev =» Eh while Ev > Eh in EBT-S. 2. The central equipotential contour lies near the machine center at the T-C transition. 3. Potential contours are nested in the T-mode and are shifted inward from the machine center. 4. The separatrix of the potential is formed in the outer region of the plasma in the direction of the major radius. For example, the separatrix 7 ORNL-DWG 84-3738 FED Fig. 2. The equipotential contours (volts) are drawn forEBT-l(f = 18 GHz, P = 50 kW, and B0 = 0.5 T) for different operating pressure. The toroidal magnetic field is directed downward into the plane of the paper in 6 B these graphs: (a) p0 = 16 X 10 torr (C-mode), (b) p0 = 14 X 10 torr (C- H 6 mode), (c) p0 = 13 X 10 torr (C-T transition), and (d) pn = 8 X 10' torr (T- mode). The measurement accuracy outside the broken circle in Fig. 2a is poor; the second cyclotron resonance zone is shown by a circle in Fig. 2d. 8 ORNL-DWG 84-3739 FED 0 X (cm) Fig. 2 (continued) 9 ORNL-DWG 84-3J40 FED -15 -10 -5 0 F 10 15 X (cm) Fig. 2 (continued) 10 ORNL-DWG 84-3741 FED Fig. 2 (continued) 11 is represented by the potential contour whose value is between 125 V and 100 V in Fig. 2d. 5. The inner potential maximum is higher than the outer potential maximum in the direction of the major radius. These general tendencies do not vary even if the microwave power is changed. We also found no differences in potential structure (Figs. 2d and 6a) when the microwave frequency was changed. It is important to note that the microwave power is launched from the top of the cavity in EBT-1 (18 GHz) and from the bottom inside (45°) in EBT-S (28 GHz), which shows that the inward shift does not result from the asymmetry of the microwave launching system. In Fig. 3, the horizontal potential profile is plotted in the T-mode of EBT-S. Near the center, the potential profile is parabolic and is approximately represented by 2 (1) 3 the electron density ng (cm- ), the central electron temperature Tg (eV), and the electric field E (V/cm) are plotted in Fig. 4 as a function of the ambient pressure for P^ = 100 kW and B0 = 0.725 T. The ion temperature is less than 20 eV. In this paper the contribution from ions to the plasma pressure is neglected. The same quantities are also plotted in Fig. 5 as a function of the 6 magnetic field with p0 = 8 X 10" torr and P^ = 100 kW. These results show that the displacement is inward from the machine center but outward from the magnetic axis (the center of the bumpy field). 12 ORNL-DWG 84C-3742 FED 250 0 - 6(x - 2.5)2 - 78 PM - 100 kW 200 1 x 10-5 torr 0.726 T 150 100 O 50 Q. -50 CLOSED POTENTIAL CONTOUR -100 x (cm) Fig. 3. The potential profile on the horizontal axis (Z = 0) is plotted in the T-mode at EBT-S (f = 28 GHz, P =100 kW , B0 = 0.725 T and 5 2 pfi = 10' torr). The solid line represents the relation $ (V) = 6(x - 2.5) - 78. Tne region of closed potential contours is displayed at the bottom of the graph. 13 ORNL-DWG 84C-3743R FED -5 1 1 -4 P^ - 100 kW _ s B0 - 0.725 T uj-T -3 - n o E <3 -2 a -1 CO 0 1 i -50 T cj ___ -40 cca E -30 i-J0 W u|E> -20 -10 0 \ UJ gg 100 - £E<> t— a: „ oS.® lli a. is UJ E SS o uz - 111 UJ o -j a — UJ X 10 16 20 AMBIENT PRESSURE ORNL-DWG 84C-3743R FED UJ 2 uj-r o £ <3 a. w ja LLI Si »-> UJ -I UJ (Jul* uj a. & E uz - uj uj o U—J» Q X 0.5 0.6 0.7 MAGNETIC FIELD STRENGTH (T) Fig. 5. The displacement of the minimum potential position from the machine center, the electric field strength at X = 5 cm, the bulk electron temperature, and the electron density at X = 0 are plotted as a function of the toroidal magnetic field strength (B0) in EBT-S (f = 28 GHz, P = 6 100 kW, and p0 = 8 X 10" torr). The minus sign in xhe displacement indicates an inward shift from the machine center. Electric fields negative for potential wells and positive for potential hills. 15 A limiter is used to determine the plasma boundary and is located between the midplane and the throat of one cavity. The limiter is shaped to fit the magnetic surface whose projection to the midplane coincides with the mod-B contour of the second cyclotron resonance when the limiter is at the operational position (i.e., about 1 cm outside the magnetic field line intersecting the location of the second cyclotron resonance in the midplane). The potential contours in Figs. 6a, 6b, and 6c are measured by changing the limiter positions. Plasma conditions in Fig. 6 are P^ = 100 kW, B0 = 6 0.725 T and p0= 8 X lO torr. In Fig. 6a, the limiter is pulled out, and the magnetic field line through the limiter terminates at the cavity wall (no toroidal connection). The limiter position in Fig. 6b is at the operational position. Fig. 6c was plotted for data with the limiter moved 2 cm inside the second cyclotron resonance position. III. DISCUSSION A. Formation of closed potential contours In a torodial plasma without rotational transform, the ions drift upward and the electrons drift downward in the plane of the potential contours (see Fig. 2). Consequently, the charge in the plasma separates and an electric field is formed that is directed downward. It is clear from Fig. 2a that this phenomenon occurs in C-mode operation, so that the plasma in the C-mode is in nonequilibrium state. As a consequence, the plasma drifts outward with a velocity of about 3 X 105 cm/s (the vertical electric field Ev = 20 V/cm in EBT-S). How do potential contours evolve from potential contours in nonequilibrium state (Fig. 2a) to those in the T-mode (Fig. 2d)? We may assume that the electrons must make at least one closed drift orbit5 in order to 16 ORNL-DWG 84-3741 FED « X (cm) Fig. 6. The relationship between potential contours and the limiter are shown. The limiter positions are (a) fully out, (b) in the operation position (1 to 2 cm outside of the field line that passes through the second cyclotron harmonic), and (c) 1 to 2 cm inside the plasma from the field line that passes through the second cyclotron harmonic resonance. 17 ORNL-DWG 84-3731A FED LIMITER JUST OUTSIDE RING T EQUIVALENT LIMITER POSITION E o N SECOND CYCLOTRON RESONANCE -15 -15 -10 -5 0 5 <0 15 (« X (cm) Fig. 6 (continued) 18 ORNL-DWG 84-3732A FED Fig. 6 (continued) 19 form closed potential contours. When the poloidal vB drift velocity becomes larger than, the outward Ev x B drift velocity and the collision frequency is lower than the poloidal precession frequency, the electrons can rotate at least once around the magnetic axis. These conditions are given by T E 5— > - , (2) Wva Bo and T -j!L_ > v • (3) 2eB0RVB r where rB is the scale length of the magnetic field gradient (resulting from the bumpy field), vg is the electron collision frequency, and r is the minor radius. We can estimate Te from Eq. (2) with rB ~ 0.35 m near the machine 3 center and Ev = 2.0 X 10 V/m with the result that T, must be larger than 1.4 keV. The electron temperature derived from Eq. (3) is above 220 eV. The bulk electron temperature in the C-T transition measured by Thomson scattering (<60 eV) is much lower than the critical temperature for the electrons to close their drift orbits due to the VB drift. This suggests that a suprathermal electron population is necessary for closed drift orbits. A high-energy electron tail can be produced in an ECH plasma when electrons become collisionless. The critical temperature for the collisionless regime is obtained from Eq. (3) and is 220 eV. An electron tail population of energy higher than 200 eV was measured in EBT11 when the operation pressure was below the C-T transition pressure. Recently, the core electron distribution function has been calculated by means of a bounce-averaged 20 Fokker-Planck code that includes quasi-linear fundamental electron cyclotron diffusion, electron-electron Coulomb collision operators, radial diffusion (8.6 times larger than the neoclassical diffusion predicted by the theory), direct loss, and ionization. It was found that a high-energy tail with an energy above a few thousand electron volts can easily be produced by" ECH heating, as shown in Fig. 7. When the limiter is positioned slightly inside the second cyclotron resonance zone, the soft X-ray signal count rate at an energy of ~1 keV decreases by a factor of 3 and the closed potential contours disappear as shown in Fig. 6c. When the error field is applied (forming an escape path for the higher-energy electrons), the soft X-ray intensity decreases and closed potential contours also are not observed. These two experiments imply that the high-energy tail electrons are needed to form the closed potential contours. At the ambient pressure above the C-T transition as shown in Fig. 2b, a strong deformation of the equipotential lines appears in the lower half of the plasma cross section. The reason this deformation occurs in the lower half plane is that in that region the poloidal VB drift of the tail electrons may cancel the outward Ey x B drift perpendicular to the electric field in the C-mode. Further acceleration of the electrons increases the poloidal VB drift velocity, which causes drift orbits to close, and thus the closed potential contours may begin to form near the center of the machine, as shown in Fig. 2c. After the closed potential contours are formed, an electric field directed toward the center of these contours forces the electrons to rotate in the same direction as the VB drift, and therefore, the area of the closed potential contours increases. For confinement this drift is equivalent to the existence of 21 ORNL-DWG 85-2007 FED u I 1 ' 1 I | I | I -2 -3 — -4 \ 123 ev\ v t = 3.8 ms-. — pt= 0 -5 — Te = 65 eV 1 I I I 1 1 1 1 1 0 4 2 3 4 5 E(keV) Fig. 7. The electron distribution function calculated by a Fokker-Planck code using EBT-S conditions. An initial Maxwellian distribution function of T = 65 eV is adopted at t = 0. After t = 3.8 ms, the distribution function changes to a bi-Maxwellian distribution of Te = 123 eV and Te = 430 eV. The formation of tail particles with energy of a few thousand electron volts is clearly evident. 22 a rotational transform. The resultant enhancement of the confinement is represented by a density increment rom the C-T transition to the T-mode, as well as a continuous temperature rise (shown in Fig. 4). Typical equipotential contours in the T-mode are shown in Figs. 2dand6a. B. Inward displacement of potential contours Let us discuss the detail of potential contours measured in EBT. In the low-density case, Popkov5 solved the Poisson equation for a simple torus. The resultant Poisson equation is similar to the Grad-Shafranov equation. Thus the potential contours are analogous to the flux functions in tokamaks. With analogy to the Grad-Shafranov equation, El-Nadi7 derived the potential formula in EBT, written as 2 , r r 2,+ a 2 — cos 0 + * 4 4 (4) where a is the ring radius, 9 is a poloidal angle, pE is an electrostatic beta value ((3K = ne(Te + T^/^ce"). and rB is the scale length of the magnetic 2 "bumpiness" (rrB = 250 cm ). The last term represents the "bumpiness." difference case of a potential well and a bell-shaped plasma pressure but that 1 <0 and f(pE) < 0 in the case of a bell-shaped potential and plasma pressure. In EBT, f(pE) is written by: a 2 dp-, L 4e where p is the plasma pressure. Generally dp/d<$> in EBT is negative. It is important that Eq. (4) indicates the inward shift of the potential contours. 23 El-Nadi simplified Eq. (5) to 8{T + T ) e l (6(b); npb) - 2 2 . MQ a where M is the ion mass, ft = 2$j/Ba2 is the radially averaged poloidal precession frequency, and g is defined from the density profile n = 2 2 n0(l - g r /a ). In EBT, the beta of the hot electron annulus is not strong enough to stabilize the interchange mode. If the plasma pressure profile is flatter than ( larger than 0.25. In Fig. 8, solutions for d<}>/dr = 0 are plotted aginst f(pE) with rB as a parameter. When we put the plasma parameters into Eq. (6), f(pE) « 7.2 for B = 0.5 T, = 250 V, and Te + T. = 30 eV for EBT-1; for EBT-S, f(|iE) « 24 for B = 0.7 T, 4>! = 350 V, and T, + T( = 100 eV, where we assume rB = 35 cm. So the theoretical inward displacement of the potential minimum with respect to the machine center is more than 4.5 cm in EBT-1 and 7 cm in EBT-S, or 3 cm in EBT-1 and 5.5 cm in EBT-S with respect to the ring center, and 1.5 cm in EBT-1 and 4 cm in EBT-S with respect to the magnetic axis as shown in Fig. 8. The measured displacements are plotted with other parameters in Fig. 4 for a pressure scan. The electron temperature increases from 60 eV (T-C transition) to 90 eV (T-M transition), and thus a larger inward displacement is predicted by theory at the T-M transition than at the T-C transition. Experimental displacements are almost constant. For the magnetic field scan, the displacements are measured as shown in 2 Fig. 5. When a is assumed to be propotional to B, f(pE) in Eq. (6) may be rewritten by 24 ORNL-DWG 85-2005R FED INSIDE HORIZONTAL POSITION 1cm) OUTSIDE Fig. 8. Solutions of d 2 /•({Ufe = g(T e +T.)/4M•i E VB (') When the magnetic field is changed from 0.58 T to 0.725 T, f(pE) increases by 3 times. Thus the inward shift should increase as B decreases in contradiction to the measurement. At B < 0.55 T, it is very difficult to estimate f(pE) because a is very small. We now compare the theoretical and experimental shifts of the potential minimum in the case of a pressure scan. The experimental data indicate an outward shift with respect to the magnetic axis, a small inward shift with respect to the ring center, and an inward shift of 2 to 3 cm with respect to the machine center as shown in Fig. 4. In the case of the magnetic field scan, the shift between 0.53 T and 0.67 T is slightly inward or an outward shift with respect to the ring center. When the machine center is chosen as the reference of the displacement, the experiment might agree qualitatively with the theory in terms of the inward shift of the potential minimum. Several experiments indicate disagreement with the theoretical predictions. 1. When an error field is applied or the limiter is positioned inside the second cyclotron resonance (Fig. 6c), the nested potential contours disappear. However, the bulk electron density and temperature do not change; that is, these plasma parameters do not depend on the potential structure. 2. The displacement of the potential contours should be shifted inward with respect to the ring center.7 The displacement of Fig. 6b is less than 1.5 cm, indicating that the potential contours are shifted outward with respect to the ring center. 26 These results indicate that the potential contours predicted by the theory are not confirmed by experimental measurements. There are several possibilities for this disagreement. First of all, in theory,7 $ is constant on the mod-B contour of the second cyclotron resonance. Experimentally, it is not constant and there is a horizontal electric field as shown iii Figs. 2d and 6b. Second, when a horizontal electric field is formed in the plasma, it cancels the concentric electric field locally where the separatrix of the potential contour is formed. Consequently, electric field convection drives the plasma out quickly-. Third, the plasma may be marginally stable at best because of a lack of reversal of <£d€/B. So, it may not be possible to apply an ideal theory to describe the EBT plasma properties. As we discuss in Sec. III.D, the electric field between the plasma and a conducting wall is increased by the inward shift of a charged plasma column. This enhanced electric field may balance the outward expansion force, which suggests that the equilibrium in the throat region may be very important because that is where the conducting wall is closest to the plasma (Fig. 1). If the equilibrium in EBT is determined in the throat region, then we can reasonably choose the machine center as r = 0. In this configuration, all of the experimental displacements are inward. With this assumption, we conclude that the closed potential contours are shifted inward from the center of the metallic boundary (machine center). C. Electrostatic beta limit Under conditions where the electrostatic equilibrium can be expressed by Eq. (4), the separatrix of the potential contours is obtained by setting d<$>/dr equal to 0. Theorists6,7 predict that the electrostatic beta value defined by Eqs. (5) and (7) should be limited when a separatrix is formed in the plasma. 27 If this is the case, the plasma in T-mode should be subject to this limitation, because a potential separatrix is experimentally found as shown in Figs. 2d, 6a, and 6b. The condition for the separatrix formation is given by7 (8) When we substitute the plasma parameters in this formula, we find that T, + T should be less than 43 eV in EBT-S. The maximum value of T, + T measured in EBT-S is about 2.5 times Larger than this calculated value. The following experiments imply that the predicted temperature limit (electrostatic P limit) is not consistent with experimental results. 1. When the error field is applied, the nested potential contours disappear, but the core temperature measured by Thomson scattering does not substantially change. 2. When the limiter is placed at 1 to 2 cm inside from the second cyclotron harmonic resonance zone, the nested potential contours disappear, as shown in Fig. 6c, but the temperature does not change substantially. 3. When the magnetic field is reduced to B = 0.58 T, there are no nested potential contours, but the electron temperature does not change or increase slightly, as shown in Fig. 5. 4. When the ambient pressure is reduced, the electron temperature increases, while a separatrix is always formed in the T-mode. These experiments indicate that separatrix formation or the structure of potential contours is independent of electron temperature. The position of the maximum potential is always close to the second cyclotron resonance zone. The nested potential structure is determined by the high-energy tail population, rather than by the electrostatic equilibrium, which is not included 28 in the theory. Thus the bulk plasma parameters do not strongly depend on the potential structure. We do not imply that the separatrix is not important. As is discussed in part E, the reason a separatrix forms in EBT is different from that predicted by theory. Once a potential separatrix is formed in the plasma, convective loss becomes dominant. Two-thirds of the input power is lost locally into the wall (according to our measurements) without traveling around the torus. D. Force balance We have examined the potential contours and the plasma parameters from the electrostatic equilibrium standpoint. We have not yet discussed the force required to counteract the 1/R outward toroidal expansion force. Popkov suggested the necessity for an external electrostatic field in the horizontal direction. This horizontal electric field in EBT is equivalent to the vertical magnetic field for tokamaks. Let us consider the electrostatic force balance. Figure 9 shows the schematic configuration. The general idea of this calculation is based on that of Bulyginskii.5 The major radius is R(m), the plasma radius is a(m), the distance between the conducting wall and the plasma is d(m), the length of the effective charge is f(m), and the dielectric coefficient in the surface 12 2 plasma is e0ex, where e0 = 8.85 X 10 F/mande » ±1.89 X lO^n/B . The expansion (ballooning) force Fb per unit length (Newton/m) is represented by (9) 29 ORNL-DWG 85-2006R FED ® B MAJOR RADIUS: R CONDUCTING WALL Fig. 9. Schematic of the electrostatic force balance. 30 where the density and the temperature are measured at the machine center. This force is directed outward. As shown in Fig. 3, a larger positive potential hill (10) The same kind of force due to Putting Fb = Fes, the necessary potential height is easily calculated, and it is written by (11) where the density of the surface plasma outside the ring is assumed to be y ne(y is the surface to core density ratio). In the case of the pressure scan shown in Fig. 4, a = 0.1 m, d = 0.015 m, e = 0.1 m, and B = 1 T, so Eq. (11) can be rewritten as (12) e The potentials calculated using ne and Te (given in Fig. 4) and y = 0.25 are compared with the measured data in Fig. 10, where the closed circles are calculated and open ones are measured, and show in good agreement. When 31 ORNL-DWG 84-3741 FED 400 ui 100 fea. 10 15 AMBIENT PRESSURE (X «)~6 lorr) 400 0.6 0.7 MAGNETIC FIELD STRENGTH (T) Fig. 10. Potentials calculated by Eq. (11) are compared with measurements. The closed circles are the calculated values. 32 we consider that y increases in the higher-pressure region, the tendency fits very well. In the case of the magnetic field scan, the quantities a and d are measured experimentally. We .assume y is constant and € is proportional to a. Using the data of Fig. 5, the calculated potential heights and the measured potential heights are compared in Fig. 10. We pointed out that an error of 20% was introduced by neglecting the contribution of the outward force. Another error comes from the measurement of d and € in Eq. (11). The total error is expected to be within about 50%, and within these error limits, it is clear that the heights are in good agreement with each other. The force balance has been studied by Freidberg8 from the toroidal magnetohydrodynamic (MHD) equilibrium point of view. He calculated the force that counteracts the 1/R outward toroidal expansion force and found that it occurs because of small helical sideband currents induced when the plasma is shifted inward with respect to the center of the bumpy field. Using EBT machine parameters to calculate the inward shift given by Freidberg, we find that the pressure center should be shifted inward by 6.6 cm from the magnetic axis. The center of the measured pressure profile is between the machine center and the magnetic axis. As pointed out by Freidberg,8 the expansion parameter in the theory is much smaller than 1, while the value for EBT is about 0.3. 33 E. Explanation of Potential Formation We have not previously discussed the mechanisms of potential formation, although we have made extensive use of experimental measurements. Generally, the outer positive portions of the potential structure are formed due to ambipolar confinement on field lines that intersect the cavity walls. This positive potential structure continues to inside the annulus on field lines that are closed. The mirror-confined electrons bounce many times in the bumpy field. The poloidal VB drifts near the midplane and in the mirror region are in the opposite directions, so the poloidal VB precession disappears somewhere along a field line. Here the particles drift vertically with a velocity of T/qRB due to lack of a rotational transform where q is the charge. Such a particle loss 12 process is called "direct loss." The warm electrons (Tw ~ 1 keV) have a larger v|(/v spread in the velocity space and are able to reach the direct loss region. The warm electrons are lost more quickly than the colder ions, causing the potential to be positive in this region. Note that the hot electrons (Th ~ 400 keV) of the annulus are confined in a narrow region of the velocity space (v|(/v ~ 0) and so are well confined in the midplane of the mirror field. This explanation is supported by the following observations: when the limiter is placed inside the annulus and an error field is applied, the soft X- ray intensity (~keV) decreases by three times and the potential well disappears, but the hard X-ray signal (> 100 keV) does not change. Details of the potential structure in this region, and particularly of the separatrix formation, can be explained by the combined effects of the horizontal and radial electric fields. For example, the separatrix of the potential contours is between 150 and 175 V in Fig. 6a. As mentioned in Sec. III.D, a horizontal electric field (Eh) due to the inward shift of the potential contours is needed to maintain electrostatic equilibrium. The radial electric field in the potential well (Er ~ 40 V/cm) is larger than Eh ~5 V/cm. However, Er near the potential hill on the inside (small major radius) of the plasma becomes comparable with the horizontal electric field. Thus these fields tend to locally cancel each other, with the resulting loss of E x B confining drifts and the degradation of particle confinement. In the T-mode, potential wells are observed to occur inside of the positive potential hill region as shown in Fig. 6b. As mentioned in Sec. III.A, closed potential contours begin to form at the T-C transition where electrons (~1 keV) can complete VB drift orbits, overcoming the outward E x B drift due to nonequilibrium electric fields. Closed drift orbits also require the presence of closed potential contours. The bulk electrons (Tg ^ 100 eV) are confined by the E x B poloidal drift, which is in the same direction for electrons as the VB drift. The reasons for the formation of a potential well are not well understood. Ion VB and E x B drifts are in opposite directions, thus tending to cancel each other with a resulting loss of ion confinement (negative potential well). However, the ions (T{ s 20 eV) will remain confined in the potential wells that are ~ 300 V deep. Ions in the tail distribution (energy < 100 eV) can escape from this potential well, but calculations indicate13 that there are insufficient numbers of these ions to maintain the observed potentials. However, the existence of an ion tail distribution suggests the presence of fluctuation potentials that can both produce the energetic ions in the tail and cause momentary decreases in the potential well depth, allowing ions to escape. We have not discussed the elongation of potential contours. It is obvious from Figs. 2d and 6a that the potential contour is not circular but is elongated 35 in the up-down direction. It may be noted that the potential contours are strongly affected by the boundary conditions (Fig. 6). When the limiter is at the operational position, the closed potential contour becomes symmetric (Fig. 6b). The nested potential contour disappears (Fig. 6c) when the limiter moves inside the second cyclotron resonance. The electric field strength Er at different radii is plotted in Fig. 11 as a function of PQ/VP where P changes from 50 kW to 200 kW and P varies from 4 X 10 6 torr to 1.5 X 10 5 torr. Near P /VP„ ~ 7 X 10 7 torr/VkW, E„ at 8 cm Or* l decreases but Er at 5.5 cm remains unchanged. The elongation results from the weakness of Er near r = 8 cm. When the symmetric potential is formed, as shown in Fig. 6b, the electric field strengths at r = 5.5 cm and 8 cm change together. Such a symmetrization of potential contours is denoted by the arrow where the limiter is located at the operational position. Fluctuations in the 150-kHz range were observed to reduce dramatically.14 These fluctuations were measured at two cavities away from the limiter. There is another experiment in which similar results were observed when additional microwave power was applied.15 Additional microwave power at a frequency of 8.5 GHz was applied at almost the same point as the second harmonic of the main frequency (18 GHz), and the hot electron stored energy increased, while the fluctuations decreased. This result is interesting because the hot electrons may prevent the penetration of the instability from outside to inside across the hot electron annuius. These two experiments suggest that (1) the fluctuations deform the concentric potential contours and (2) the main source of this fluctuation is not located inside but outside the hot electron ring position. The E x B drift frequency coE x B is calculated as 36 ORNL-DWG 84-3070R3 FED Po'SPy. U10"7torrA/kW) Fig. 11. The electric field strengths at r = 5.5 and 8 cm are plotted as a function of PQ/VP for microwave powers in the range 50 to ^200 kW. The heavy-ion-beam trajectory and local positions are shown in Fig. 6a. The arrow in the figure represents the symmetrization of the potential contour when the limiter moves to the operation position. 37 u,ExB= "i7~ = <0.8-1.2) X lO^s"1) (13) 4 1 The electron-drift frequency coe* due to the density gradient is 10 s" and is 16 much smaller than ioEx B. Komori measured a fluctuation with a mode number (m) of 10 in the same range by a Langmuir probe. He identified this as a flute mode because this fluctuation rotates in the ion drift direction. If 6 1 the fluctuation measured here is assumed to be m = 10, then m ooE x B = 10 s and is very close to the measured frequency 2nf = 9 X 105s"'. IV. CONCLUSION The equilibrium in EBT has been studied. The potential structure in the C-mode represents a lack of equilibrium. At the T-C transition, nested potential contours start to form. Warm electrons (~keV) are necessary for the formation of closed potential contours. : Experimental results have been compared with theories from the viewpoint of the inward shift of the potential well and the electrostatic beta limit (separatrix formation). The experimental inward shift is consistent with the theory5-6-7 when allowance is made for the effect of the walls in the mirror throat. An electrostatic 0 limit has not been found. One of the main difficulties in comparing experiment with theory6-7 is that the formation of the separatrix is found to be independent of core plasma parameters, which results from the cancellation of the weak electric field near the annulus with the horizontal electric field in the experiment. The MHD equilibrium has also been discussed. The ideal theory may not be applicable to the EBT plasma because of differences between experimental conditions and the theoretical model. 38 The electrostatic force balance has been suggested to cancel the toroidal expansion force. When the charge column shifts inward, the conducting wall surrounding the plasma induces an image charge and consequently pulls the plasma inward. This force balances the toroidal expansion force (nT/R). A simple estimation of this balance force is in good agreement with observations. 39 jH ® ACKNOWLEDGMENTS The authors express their appreciation to R. E. Juhala and D. W. Swain for use of the limiter. Thanks are also extended to the members of the EBT group and the EBT operation group for their support. One of the authors (Hiroe) would like to express his appreciation to B. H. Quon (Jaycor, Torrance, California) for his discussion and to M. Fujiwara (Institute of Plasma Physics, Nagoya University, Japan) for his introduction to this problem. 41 REFERENCES 1. R. A. Dandl, R. A. Dory, H. O. Eason, G. E. Guest, C. L. Hedrick, H. Ikegami, and D. B. Nelson, in Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Tokyo, 1974), Vol. 2, p. 141. R. J. Colchin, T. Uckan, F. W. Baity, L. A. Berry, F. M. Bieniosek, L. Bighel, W.A.Davis, E.Dullni, H. O. Eason, J. C. Glowienka, G.A. Hallock, G. R. Haste, D. L. Hillis, A. Komori, T. L. Owens, R. K. Richards, L. Solensten, T. L. White, and J. B. Wilgen, Plasma Phys. 25,597 (1983) (and references listed therein). 2. G. I. Budker, in Plasma Physics and the Problem of Controlled Thermonuclear Reactions, edited by M. A. Leontovich (Pergamon, New York, 1971) Vol. l,p. 78. 3. T. H. Stix, Phys. Fluids 14, 692 (1971). 4. J. D. Daugherty and R. H. Levy, Phys. Fluids 10,155 (1967). 5. D. G. Bulyginskii, V. S: Yuferev, and E. V. Galaktinov, Sov. J. Plasma Phys. 3(5), 529 (1977). 6. N. G. Popkov, Sov. J. Plasma Phys. 5(4), 482 (1979). 7. A. M. El-Nadi, Phys. Fluids 28, 878 (1985). 8. J. P. Freidberg, Nucl. Fusion 20, 673 (1980). 9. P. L. Colestock, K. A. Connor, R. L. Hickok, and R. A. Dandl, Phys. Rev. Lett. 40,1717 (1978); F. M. Bieniosek, and K. A. Connor, Phys. Fluids 26, 2256(1983). 10. S. Hiroe, J. B. Wilgen, F. W. Baity, L. A. Berry, R. J. Colchin, W.A.Davis, A.M. El-Nadi, G.R. Haste, D.L. Hillis, T.L.Owens, D. A. Spong, and T. Uckan, Phys. Fluids 27,1019 (1984). 11. D. L. Hillis, G. R. Haste, and L. A. Berry, Phys. Fluids 26,820 (1983). 42 12. D. A. Batchelor, EBT Experimental Group, and EBT Theory Group, Proceedings of the 4th International Symposium on Heating in Toroidal Plasma (International School of Plasma Physics, Varenna, 29284), Vol.11, p779. 13. E. F. Jaeger, D. E. Hastings, C. L. Hedrick, and J. S. Tolliver: private communication. 14. S.Hiroe, J. C. Glowienka, D. L.Hillis, J. B.Wilgen, G. L.Chen, J. A. Cobble, A. M. El-Nadi, J. R. Goyer, L. Solensten, W. H. Casson, O. E. Hankins and B. H. Quon: submitted to Phys. Fluids (Fig. 7). 15. H. Iguchi: private communication. 16. A. Komori, Nucl. Fusion 24,1173 (1984). 43 ORNL/TM-9520 Dist. Category UC-20 f,g INTERNAL DISTRIBUTION 1. F. W. Baity 24. R. K. Richards 2. D. B. Batchelor 25. J. Sheffield 3. L. A. Berry 26. D. A. Spong 4. T. S. Bigelow 27. D. W. Swain 5. W. H. Casson 28. J. S. Tolliver 6-7. G. L. Chen 29. N. A. Uckan 8-9. R. J. Colchin 30. T. Uckan 10. J. C. Glowienka 31. T. L. White 11. G. R. Haste 32. J. B. Wilgen 12. C. L. Hedrick 33-34. Laboratory Records Department 13. D. L. Hillis 35. Laboratory Records, ORNL-RC 14-18. S. Hiroe 36. Document Reference Section 19. E. F. Jaeger 37. Central Research Library 20. H. D. Kimrey 38. Fusion Energy Division Library 21. L. W. Owen 39-40. Fusion Energy Division Publications 22. T. L. Owens Office 23. D. A. Rasmussen 41. ORNL Patent Office EXTERNAL DISTRIBUTION 42. Office of the Assistant Manager for Energy Research and Development, U.S. Department of Energy, Oak Ridge Operations, P.O. Box E, Oak Ridge, TN 37831 43-44. J. A. Cobble, Los Alamos National Laboratory, Los Alamos, NM 87545 45-46. K. A. Connor, Rensselaer Polytechnic institute, 110 Eighth St., Troy, NY 12181 47-48. J. R. Goyer, Rensselaer Polytechnic Institute, 110 Eighth St., Troy, NY 12181 49-50. L. Solensten, Rensselaer Polytechnic Institute, 110 Eighth St., Troy, NY 12181 51. J. M. Turner, Mirror Systems Branch, Office of Fusion Energy, Office of Energy Research, Mail Stop G-256, U.S. Department of Energy, Washington, DC 20545 52. M. Fujiwara, Institute of Plasma Physics, Nagoya University, Nagoya 464, Japan 53. T. V. George, Office of Fusion Energy, Office of Energy Research, Mail Station G-256, U.S. Department of Energy, Washington, DC 20545 54. T. Shoji, Institute of Plasma Physics, Nagoya University, Nagoya 464, Japan 55. A. Komori, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816, Japan 56. N. A. Krall, JAYCOR, 11011 Torreyana Rd., P.O. Box 85154, San Diego, CA 92138 44 57. S. Hamasaki, JAYCOR, 11011 Torreyana Rd., P.O. Box 85154, San Diego, CA 92138 58. N. H. Lazar, TRW Defense and Space Systems, 1 Space Park, Bldg. R-l, Redondo Beach, CA 92078 59. J. B. McBride, Science Applications, Inc., 1200 Prospect St., P.O. Box 2351, La Jolla, CA 92037 60. W. B. Ard, McDonnell Douglas Astronautics Company, Bldg. 278, P.O. Box 516, St. Louis, MO 63166 61. H. L. Berk, Institute of Fusion Studies, University of Texas at Austin, Robert L. Moore Hall, Rm. 11.218, Austin, TX 78712 62. F. M. Bieniosek, McDonnell Douglas Astronautics Company, Bldg. 278, P.O. Box 516, St. Louis, MO 63166 63. R. A. Dandl, Applied Microwave Plasma Concepts, 2210 Encinitas Blvd., Suite P, Encinitas, CA 92024 64. R. S. Post, Plasma Fusion Center, Massachusetts Institute of Technology, 190 Albany St., Cambridge, MA 02139 65. D. H. McNeill, Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, NJ 08544 66. K. Mizuno, Applied Science Dept., University of California—Davis, 228 Walker Hall, Davis, CA 95617 67. F. L. Ribe, College of Engineering, AERL Building, FL-10, University of Washington, Seattle, WA 98195 68. T. C. Simonen, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 69. H. Weitzner, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 70. K. Tsang, Science Applications, Inc., 934 Pearl St., Boulder, CO 80302 71. B. Fried, Department of Physics, University of California, Los Angeles, CA 90024 72. J. W. Van Dam, Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712 73. A. Wong, Department of Physics, University of California, Los Angeles, CA 90024 74. H. Goede, R1/2144, TRW Inc., Energy Development Group, One Space Park, Redondo Beach, CA 90278 75. H. K. Forsen, Bechtel Group, Inc., Research Engineering, P.O. Box 3965, San Francisco, CA 94105 76. J. R. Gilleland, GA Technologies, Inc., Fusion and Advanced Technology, P.O. Box 81608, San Diego, CA 92138 77. R. A. Gross, Plasma Research Laboratory, Columbia University, New York, NY 10027 78. D. M. Meade, Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08544 79. F. Prevot, CEN/CADARACHE, Department de Recherches sur la Fusion Controllee, 13108 Saint-Paul-Lez-Durance, Cedex, France 80. M. Roberts, International Programs, Office of Fusion Energy, Office of Energy Research, ER-52 Germantown, U.S. Department of Energy, Washington, DC 20S45 45 81. W. M. Stacey, School of Nuclear Engineering, Georgia Institute of Technology, Atlanta, GA 30332 82. D. Steiner, Rensselaer Polytechnic Institute, Nuclear Engineering Department, NES Building, Tibbets Avenue, Troy, NY 12181 83. R. Varma, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India 84. R. W. Conn, Department of Chemical, Nuclear, and Thermal Engineering, University of California, Los Angeles, CA 90024 85. J. D. Callen, Department of Nuclear Engineering, University of Wisconsin, Madison, WI 53706 86. D. D. Ryutov, Institute of Nuclear Physics, Siberian Branch of the Academy of Sciences of the U.S.S.R., Sovetskaya St. 5, 630090 Novosibirsk, U.S.S.R. 87. S. O. Dean, Director, Fusion Energy Development, Science Applications, Inc., 2 Professional Drive, Gaithersburg, MD 20760 88. G. A. Eliseev, I. V. Kurchatov Institute of Atomic Energy, P.O. Box 3402, 123182 Moscow, U.S.S.R. 89. V. A. Glukhikh, Scientific-Research Institute of Electro-Physical Apparatus, 188631 Leningrad, U.S.S.R. 90. N. A. Davies, Office of Fusion Energy, Office of Energy Research, Mail Stop G-256, U.S. Department of Energy, Washington, DC 20545 91. G. Gibson, Westinghouse Electric Corporation, Fusion Power Systems Department, P.O. Box 10864, Pittsburgh, PA 15236 92. R. W. Gould, Department of Applied Physics, California Institute of Technology, Pasadena, CA 91109 93. D. G. McAlees, Exxon Nuclear Company* Inc., 2101 Horn Rapids Road, Richland, WA 99352 94. J. F. Clarke, Associate Director for Fusion Energy, Office of Energy Research, Office of Fusion Energy, Department of Energy, Mail Station G-256, Washington, DC 20545 95. D. B. Nelson, Acting Director, Division of Applied Plasma Physics, Office of Fusion Energy, Office of Energy Research, Mail Stop G-256, U.S. Department of Energy, Washington, DC 20545 96. Documentation S.I.G.N., Departement de la Physique du Plasma et de la Fusion Controlee, Centre d'Etudes Nucleaires, B.P. No. 85, Centre du Tri, 38041 Cedex, Grenoble, France 97. Bibliotheque, Service du Confinement des Plasmas, CEA, B.P. 6, 92 Fontenay-aux-Roses (Seine), France 98. Bibliothek, Institut fur Plasmaphysik, KFA, Postfach 1913, D-5170 Julich, Federal Republic of Germany 99. Bibliothek, Max-Planck Institut fur Plasmaphysik, D-8046 Garching bei Munchen, Federal Republic of Germany 100. Bibliotheque, Centre de Recherches en Physique des Plasmas, 21 Avenue des Bains, 1007 Lausanne, Switzerland 101. Library, Culham Laboratory, UKAEA, Abingdon, Oxfordshire, OX14 3DB, England 102. Library, FOM Instituut voor Plasma-Fysica, Rijnhuizen, Edisonbaan 14, 3439 MN Nieuwegein, The Netherlands 46 103. Library, Institute of Physics, Academia Sinica, Beijing, Peoples Republic of China 104. Library, Institute of Plasma Physics, Nagoya University, Nagoya 64, Japan 105. Library, International Centre for Theoretical Physics, Trieste, Italy 106. Library, Laboratorio Gas Ionizzati, 1-00044 Frascati, Italy 107. Library, Plasma Physics Laboratory, Kyoto University, Gokasho, Uji, Kyoto, Japan 108. Plasma Research Laboratory, Australian National University, P.O. Box 4, Canberra, A.C.T. 2000, Australia 109. Thermonuclear Library, Japan Atomic Energy Research Institute, Tokai, Naka, Ibaraki, Japan 110-301. Given distribution as shown in TID-4500, Magnetic Fusion Energy (Category UC-20 f,g: Theoretical Plasma Physics and Experimental Plasma Physics) • U.S. GOVERNMENT PRINTING OFFICE: 1986-631 -056/40020 /E-