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P/2212 USSR Stabilization of Plasma by Nonuniform Magnetic Fields By B. B. Kadomtsev and S. I. Braginsky:| wards from the boundary. This result is confirmed by a rigorous hydrodynamic treatment. The equations of magnetohydrodynamics for small plasma oscillations can be written as a single selfconjugate differential equation of the second order, for a displacement £ from an equilibrium position. This equation can be obtained from a variational principle with a real functional. If the plasma pressure po and its density /><) are constant and the currents flow only along its surface, one obtains for the proper oscillation frequencies from the variational principle: STABILIZATION OF PLASMA BY MEANS OF GUARDING CONDUCTORS The main problem in controlled thermonuclear reactions is the confining of high-temperature plasma, i.e., the obtaining of a stable plasma configuration isolated from the walls. One of the most direct methods for creating a plasma separated from the walls consists in the application of the well-known pinch effect stabilized by a longitudinal magnetic field and a metal casing (this problem is considered in detail by S. I. Braginsky and V. D. Shafranov). It is of interest to study other possibilities of obtaining a stable plasma configuration. The present paper deals with the problem of stabilizing plasma by means of a nonuniform magnetic field of a special kind. The systems considered here can, obviously, be used both separately and in combination with the pinch effect. Let us consider a case when the magnetic field within the plasma is absent. We shall assume that the conductivity of the plasma is infinite, and all the currentsflowingalong the plasma are concentrated on its surface. The criterion of stability of a plasma such as described above can be defined by. simple qualitative considerations. Indeed, the instability of the plasma cylinder pinched by the magnetic field of its own current is closely connected with the decrease of the magnetic field, outwafds from the boundary of the plasma. This becomes particularly clear if we consider local perturbations of the boundary. If there is no magneticfieldwithin the plasma, the plasma pressure in equilibrium is balanced by the external magnetic field P = Я2/8тг, where Я is the value of the field at the boundary. Let us imagine that the plasma ejects a flat "tongue" oriented parallel to the magnetic Unes of force. Such a "tongue" slightly perturbs the magnetic field—as if it were pushing through the lines of force and slightly drawing them apart. If the field decreases outward from the plasma boundary, the tip of the "tongue" touches the region of lower magnetic pressure and will stretch out further with an acceleration. f -Л2 Jv{ yPo where Vi is the volume occupied by the plasma, Ve is the external volume, So is the plasma boundary, Hoe and Яо1 are the unperturbed fields outside and inside the plasma respectively; p and H are the pressure and field of small perturbations, f» is the normal component of the displacement on the boundary So and д/дп is a derivative along the external normal to SoThe above equation shows that the observance of the condition 8[{Н0в)2-{Н0Щдп > 0 at all points of the plasma boundary is sufficient to ensure plasma stability (to2 > 0). If Ho1 = 0, an increasingfieldaway from the plasma boundary is a sufficient and, as is seen from the previous qualitative consideration, also a necessary condition for its stability. (Similar results were obtained in Ref. 1.) Hence, we see the way to find stable systems. For instance, the infinite flat plasma layer (Fig. l(a)), outside of which the field is constant, shows no instability.2 However, a flat conductor of infinite broadness cannot be made and for any finite broadness instability appears on account of the edge effect. As is well known the edge effect in a flat condenser can be eliminated with the help of guard rings. An analogous method can be used in this case too. The system given in Fig. 1(6), in which part of the plasma is substituted by metal conductors, is stable. With increase in the magneticfieldthe plasma boundary is driven inwards, and in this case the magnetic field increases outward from the plasma boundary (Fig. l(c)). Such a system is all the more stable. Therefore, it seems natural that for stability of such a plasma, it is necessary for the field to increase outOriginal language: Russian. * Academy of Sciences of the USSR, Moscow. Г H% J r{+ve 4тг M a[(ffOe)2-.(ffol)2] 8n dn 233 234 SESSION A-9 P/2212 B. B. KADOMTSEV and S. I. BRAGINSKY (b) 2212.1 Figure 1. Stable plasma configurations One significant fact is that the guard conductors prevent the lines of forcefrbm freely moving with the plasma. The lines of force tighten on the conductors like elastic strings and because of this no sausage-type instability and distortions of the cylindrical plasma column appear. This ejfect can be further intensified if the plasma is surrounded by a lattice of metal rods placed perpendicular to the Unes of force (Fig. l{d)). If the lattice is sufficiently dense, it rigidly determines the plasma boundary, provided the magnetic field pressure outside the,lattice is larger than the plasma pressure. Indeed, the magneticfieldcannot sag deeply through the dense lattice. Therefore, if the plasma (a) Figure 2. Isolation of plasma from rods boundary moves away from the lattice inwards appreciably, the magnetic field on it decreases drastically. If the plasma moves outwards, the current from the rods passes on to the plasma which will take up the whole magnetic pressure, greater than the plasma pressure. As a result a certain equilibrium and stable state of the plasma boundary will be established which, roughly speaking, coincides with the lattice surface. In this case part of the magnetic pressure is held by the rods and stability is reached at the cost of some increase in the value of the magnetic field increase balancing the plasma. If the lattice is flat, it is possible to consider in detail the plasma boundary near the rods. Since outside the plasma and rods, curl H = 0 and div H = 0, and Я 2 = 8TT/> = const on the plasma boundary, the problem of the plasma boundary is analogous to the hydrodynamic problem of potential flow of an incompressible fluid with free boundaries. By known methods of conformai transformations, it is possible to find the magnetic field and the shape of the plasma boundary. If the magnetic field pressure does not exceed the plasma pressure too much, the plasma boundary sags between the rods, as is shown in Fig. 2(Й). In this case the magnetic field increases in the outward direction from the plasma boundary, so that it is stable with respect to small perturbations. If there is some magnetic flux between the rods and the plasma, the plasma boundary takes the position illustrated in Fig. 2(6). In this case the plasma never touches the rods, and there is no direct heat exchange between the plasma and the rods. Such a configuration is also stable with respect to small perturbations. Thus, the guard lattice gives the plasma boundary stability with respect to small perturbations and also leads to a constant stabilizing force for displacements which are large in comparison with the distance between the rods. The guard conductor systems can differ widely. The simplest system is the plasma cylinder; the lattice can be made either of straight rods arranged along the generatrix of the cylinder or of rings perpendicular to the generatrix. In the first case (Fig. 1(¿)) the current in the plasma and in the rods flows in the direction of the cylinder axis. This current produces an azimuthal magnetic field. In the ring system the longitudinal magnetic field is produced by an additional external winding. The azimuthal current flows along the rings and the plasma in a direction reverse to the direction of the current in the winding. Each of these systems can be either transformed into a torus or restricted at the ends. In the rod system, for instance, electrodes can be placed at the ends, and the ring system can be restricted by intensifying the magnetic field near the ends ("magnetic plug") with a simultaneous decrease in the diameter of the extreme rings (Fig. 3). In the last system the presence of a longitudinal magnetic field within the plasma is very undesirable since it facilitates the release of the plasma from the system. PLASMA STABILIZATION The guard conductor systems are, generally speaking, not stationary. Their time of action is restricted by the finite value of electrical conductivity of the plasma, as well as of the conductors if the- latter are passive, i.e., if they are not fed by energy frorri'the external circuit. Indeed, the entire plasma will move beyond the lattice, where it will be released because of the loss of stability, during the time tB — 4iro-a2/c2, where a is the transverse plasma dimension and a is its conductivity. In reality the lifetime of the plasma may be significantly less. For instance, in the system illustrated in Fig. 3, with a sufficiently large frequency of collisions, the plasma in the skin layer will escape along the lines of force with a thermal ion velocity VT- If the length of the system equals L, the time of escape of particles along the lines of force is ir = L/VT. As a result of the plasma loss from the skin layer a certain thickness of the skin layer independent of the time is established, f and the plasma lifetime t will become, obviously, L\i With sufficiently high plasma conductivity t ^ ta. A similar situation is found in other systems, viz., as a result of diffusion of the magneticfield,the plasma moves beyond the rod to the line ABC in the solution presented in Fig. 2(a) or to the dotted line of force in the isolated solution presented in Fig. 2(b), so that around each rod there appears a plasma "encasement". But each rod must somehow be suspended from the walls of the chamber; the supports would pass through this plasma "encasement". Consequently, in the system there should occur losses on the supports. If the supports have a diameter b, and the distance between the supports of a given rod is /, the probability for the particle to escape when passing through one "encasement" equals b/l. Hence, the lifetime of particles in the skin layer is tT = L/VTJ/Ь, where L ~ a is the distance between the rods. By repeating the above calculations we obtain , ..... '№)* /4яО«» L I \* ( j which, at b/l ~ 10~2, is one order greater than the lifetime of the system presented in Fig. 3. The supports submerged in hot plasma are subject to a powerful plasma ion bombardment which results in their evaporation and in contamination of the system. The evaporation of the supports can be considerably decreased if they are supplied with a larger current, so that the magnetic Unes of force near the supports close. Then the ions can reach the supports only by t The mean skin layer thickness Larmor radius of electrons if 8 < />e. should be taken as a 235 Winding 2212.3 Figure 3. Ring and magnetic plug geometry diffusion, and the density of the energy loss will be of the order of Tnp2 -г-— О T where n is the plasma density, T is its temperature, p is the mean Larmor ion radius, т is the ion-ion collision time and w = eH/Mc is the ion cyclotron frequency. If p/b ~ 1, the energy loss decreases от times in comparison with the energy release nTv^ on the currentless support. In this case there appears along the supports at some distance from it a region of very weak magnetic field which will play thé part of a "canal" for the plasma leakage. But if the width of the canal is of the order of b\ this leakage, in any case, will not exceed the direct loss to the supports. It follows from the estimates given that though the plasma lifetime is less than the skin-time, it can reach several hundredths of a second at a ~ 102 and T of the order of several kev. The estimates show that for a D-T mixture the thermonuclear reaction energy released in this time can reach the value of the thermal energy. It may be expected that the most promising is the use of the guarding conductors in combination with the pinch effect. As is known from Ref. 3, the plasma pinch in the toroidal system possessing an azimuthal symmetry can be in equilibrium only in the presence of a current along the pinch. Therefore, alternation of the current is not permissible in such a system: at the moment of zero current equilibrium is lost. If a torus with a strong magnetic field is supplemented by guard rings, the plasma at the moment of zero current will be retained by the rings,' and this will make it possible to extend the discharge over many periods of the axial current. On the other hand the self field of the current compressing the pinch apparently leads to a decrease in loss on the suspenders. These problems require experimental investigation. MAGNETIC TRAPS The problem of plasma stabilization may be approached by a different, in some sense opposite, method. The condition for stability obviously becomes more and more severe with, an increase in plasma pressure. Therefore, if we consider the case of a plasma 236 SESSION A-9 P/2212 B. B. KADOMTSEV and S. I. BRAGINSKY Both in closed-line traps and in adiabatic traps the particles travel quasi-periodically; they move mainly along the Unes of force, gradually drifting from one Une to another. If the magnetic field is strong enough so that the total drift during one passage of the Une of force is small, then during such a movement the longitudinal invariant / = \udl, as can be shown, is conserved (u is the longitudinal velocity component, <п is the element of length of the Une of force; the drift velocity itself is proportional to the gradient J). If the electricfieldis absent, then v = const, and the equation for the surface along which the particle drifts can be written in the form Figure 4 with a pressure much lower than that of the magnetic field, it will enable us to determine the minimum requirements for the field which is capable of confining the plasma. Under such conditions it is a property of the field itself, since the low-pressure plasma plays only the part of a "test body"— practically, it does not distort the field under investigation. A field capable of confining a quasi-neutral plasma with an infinitesimal pressure will be called a magnetic plasma trap. Consideration of magnetic traps makes it possible to obtain a simple plasma confinement criterion which provides some orientation in more complicated cases. Magnetic Particle Traps Since plasma confinement implies the confinement of separate plasma particles and since the investigations of the behaviour of charged particles can often predict the behaviour of plasma, it would be of interest to investigate the simpler problem of magnetic particle traps, i.e., magnetic fields that can hold charged particles in a limited space for a long time. | We shall start our consideration with these particle traps. Particle tiaps naturally fall into two groups: traps with closed Unes of force, and adiabatic traps based on the conservation of the transverse invariant I = œ>2/ff, where w is the velocity component perpendicular to the magnetic field. Only those particles which experience a reflection from the "plugs", i.e., from the strong magneticfieldregions, are confined in the adiabatic traps suggested by G. I. Budker. Particles with a small w/v ratio pass out through the "plugs" and one would like by some means to return them. However, an ordinary closing of the lines of force will not bring us to our aim: the particles moving in a closed line will gradually drift to the walls. An analogous drift takes place in all closed-line traps, and the problem of confining the particles leads to the investigation of this drift. $ Magnetic particle traps are also of interest from the point of view of confinement of the charged products of the thermonuclear reactions. Thus, in order to keep a particle with a given v and J within a certain volume, the ф surface must be closed and remain entirely inside this volume. The condition is simplest for particles with ujv = 1, i.e., J = 0. For these particles ф simply denotes the length of the Une of force, and consequently, they will be confined if the magnetic field is "corrugated", i.e., if the peripheral Unes of force are longer than the internal Unes. As an example, we shall consider two corrugated magnetic field traps. Each of them consists of two corrugated traps with plugs coupled to one another with a "cross-link" A. The presence of plugs with field Hm will lead to the confinement of a great number of particles in the corrugated parts and only particles with w%\v% < Ho¡Hm ^ 1 will pass over the cross-link, travelling along the closed Unes of force. The first system (Fig. 4(a)), which can be called a closed magnetic plug trap, operates to a certain extent like the ordinary adiabatic trap. If L > R, the length of the Une of force is determined by the corrugated part and the share of the cross-Unk is insignificant. Therefore the particles that have passed over the cross-link will be stabilized: corrugation leads to a compulsory circular drift of particles near the axis of the system which compensates the drift in the crosslinks, providing ДХ/ТГЙ > 1, where AL is the elongation of the peripheral Une in comparison with the central Une. Therefore, the main process of particle loss in such a system is due to particle-particle colUsions in the cross-links A which wiU lead to their drifting out of the cone of wz/v2 < Ho/Hm and to a subsequent drifting out to the cross-Unk wall. But since the density in the cross-Unk is Ho/Hm times less than the density in the straight parts and their volume is TTR/L times less than the volume in the straight part, the leakage will decrease HmLIHOirR > 1 times. Magnetic Plasma Traps The second system (Fig. 4(6)) is some modification of the torus. Here the bulk of particles in the cross-links is also stabilized. However, the particles with small u/v ~ p/a in the cross-links are able to settle on the wall during the time of theirflightover the cross-links, 237 PLASMA STABILIZATION which leads to an additional energy leakage exceeding the leakage caused by thermal conductivity if A > L, where A is the mean free ion path. ' Besides, in both traps there occurs an increase in the thermal conductivity on account of the radial displacement caused by the drift in the cross-links ("mixing"). Therefore, if the plasma touches the wall, the linear part must be of a very great length to fulfil the condition LLJTTR > 1. Corrugated Traps Corrugated magnetic traps can confine individual particles, or at least most of them. But this does not mean that they will confine aquasi-neutral plasma, since there occur in the plasma proper electric fields which can essentially change the character of the individual particle motion. The next problem consists in finding conditions of confinement for a plasma with an infinitesimal pressure. In the hydrodynamic approximation and with the assumption of an infinite conductivity these conditions have a simple form and can be deduced from simple qualitative considerations. For the low-pressure plasma only such motions can exist that shift separate tubes of force almost without distorting the magnetic field. For such motions the tube volume of a tube of plasma changes proportionally to the integral / (1/H)dl along the line of force. Plasma trying to expand will induce each tube to tend towards an increase of this integral. As a result, it appears5 that the behaviour of lowpressure plasma is analogous to that of a nonuniformly heated gas in gravitational field; the plasma pressure corresponds to the gas density and the function U = —f(l/H)dl is the potential energy. If the plasma at the initial moment has an arbitrary distribution, convection sets in: the tubes of force in which the plasma pressure is higher start moving towards U, ousting the tubes with lower pressure. Thus, the plasma tends toward a stable state in which its pressure declines with increasing U. Besides this, other stable equilibrium states exist. Just as a stable equilibrium is possible in the atmosphere of the earth when the air temperature does not drop too fast with height, in a plasma there is no loss of stability if its pressure rises not very quickly with increase in U. Indeed, let the plasma be in equilibrium and let a certain tube shift towards an increase in U by some infinitesimal quantity, moving the other tubes apart. If the process occurs adiabatically, the plasma pressure in this case will change by a quantity Thus, the problem of plasma confinement by a given magnetic field requires an investigation of the potential U for this field. For a magnetic trap the potential U should be minimum within a bounded volume and increase towards the walls. It appears that the field of the guard conductor systems satisfies this condition completely. Let us consider, for example, a guard ring system the magneticfieldof which has the form presented in Fig. 5. Since the system in Fig. 5 is periodic, U may be regarded as an integral over one period, — a/2 < z < a/2. The potential U is a function of the line of force; therefore it is convenient to consider it in the plane z = 0, crossing all the lines of force. In this plane U has two "wells" at the points A and С located on the dotted line of force that passes through A, where the field vanishes. At A, U logarithmically approaches minus infinity. Near the rod U has a maximum and is close to zero. At v = 0, U = — u/Hi, \ Winding Rings Figure 5. Corrugated trap geometry where Hi is a certain mean value of the field on the z axis; and at point D, U £ —a/Ho > — a/Hi. Hence, we see that any plasma configuration whose pressure decreases in the direction away from the dotted line of force is certainly stable. Moreover, an arbitrary plasma configuration in such a system changes by itself, so that the pressure decreases in the direction away from the dotted line, i.e., the plasma itself will tend to a stable equilibrium. Besides states in which there are plasma encasewhere у is the adiabatic index. If the pressure in the ments near the rings, we can find in the ring system tube appears to be lower than the surrounding plasma a stable plasma state whose pressure differs from zero pressure, which is equal to p + dp, the tube win move only within the dotted line of force, and vanishes on with acceleration towards further increase in U; if this Une; for stability it is sufficient that the plasma 8p < dp, the tube win be forced back and the plasma pressure decreases with U slower than | U\-v. will be stable. Thus, to obtain stability it is necessary However, the actual attainment of such a state is a and sufficient that dp/dU < yp/\U\, i.e., the plasma more subtle matter; it requires a special preparation of pressure should drop with U not faster than \U\-v. the plasma. Moreover, this state actually corresponds 238 SESSION A-9 P/2212 B. B. KADOMTSEV and S. I. BRAGINSKY to the plasma touching the wall, since the state assumes that there should be constantly a pressure equal to zero on the dotted line, otherwise plasma encasements will appear. Being stable, the system presented in Fig. 5 can be transformed into a torus: for a very small toroidality there will be no loss of stability. An analogous consideration shows that there are stable low-pressure plasma states in other guard conductor systems. This means that guard conductor systems are also magnetic plasma traps. This circumstance, although not well proven, can serve as an important argument in favour of a possible plasma stabilization in guard conductor systems in a broad pressure region. The stability conditions for magnetic traps with a corrugated field follow also from the above considerations. Indeed, the magnetic field in the system presented in Fig. 5 is a field of a corrugated type, and the fact that the plasma in such a field tends to take a position along the dotted line shows that, at least for the case of plasma states separated from the wall, m'agnetic traps with corrugated fields should be practically realized as guard conductor systems. ACKNOWLEDGEMENT The authors' best thanks are due to Prof. M. A. Leontovich for his participation in the discussion of the work and for his valuable suggestions in the process of its fulfilment. REFERENCES 1. I. Bernstein, E. Frieman, M. Kruskal and R. Kulsrud, Proc. Roy. Soc, A., 224, 17 (1958). 2. M. Kruskal and M. Schwarzschild, Proc. Roy. Soc, A., 223, 348 (1954). 3. V. D. Shafranov, Zhur. Eksp. i Teoret. Fiz., 33, 710 (1957). 4. M. Rosenbluth and С Longmire, Ann. Phys., 1, 120 (1957). 5. R. Z. Sagdeev, B. B. Kadomtsev, L. I. Rudakov and A. A. Vedenov, Dynamics of Rarified Plasma in a Magnetic Field, P/2214, Vol. 31, these Proceedings.