Patent Number: 043022847
Section: description

As illustrated in FIGS. 1 and 2, a toroidal plasma device 10 includes a primary confinement vessel in the form of a toroidal liner 12 which confines and defines a primary toroidal chamber 14 containing appropriate gas at a suitable low pressure. In the design illustrated, the liner 12 is made of thin wall stainless steel which permits rapid penetration of toroidal electric field to start up and drive plasma current in the primary toroidal chamber 14. The toroidal liner 12 is disposed within and supported from a secondary confinement vessel in the form of a toroidal shell 16. The shell 16 as shown is formed of a relatively thick copper wall forming a secondary toroidal chamber 18. The secondary chamber 18 is evacuated through conduits 20 and a header 22 by a vacuum pump means not shown. The primary chamber 14 is evacuated through conduits 24 and a header 26 by vacuum pump means also not shown. As shown in FIGS. 4 and 5, the shell 16 includes a ceramic break 28 which serves to interrupt the toroidal conductive path around the shell 16 which would otherwise short circuit the toroidal conductive path through the plasma. The conductance of the liner 12 is sufficiently low in respect to the conductance of the plasma as not to be wasteful of energy. That is, a magnetic field may readily penetrate the conductive shell 16 because of the ceramic break 28 and penetrate the liner 12 because it is relatively thin and of lower conductivity than the material forming the shell 16. At the same time, the liner 12 provides an electrical bridge across the ceramic break 28 and isolates the ionized plasma from the electrical break thereby formed in the conductive shell 16. At the same time, the conducting shell 16 aids in stabilization of the plasma by repelling plasma current trying to move toward the wall of the shell 16. As with tokamak devices, the plasma current is produced by a toroidal electric field induced by a solenoid coil 30 disposed axially of the major axis of the toroidal liner 12 and inside the torus. The toroidal electric field is created by operation of the solenoid coil 30 and additional turns 32 disposed to channel the poloidal flux outside the liner 12. The solenoid coil 30 and additional turns 32 are energized in a conventional manner by a power supply not shown, whereby the change in electrical current in the coil causes a change in magnetic flux linking the single turn secondary formed by the liner 12 and its contents. The change in flux, in turn, generates plasma current within the primary chamber 14. A plurality of first windings 34 are wound substantially helically upon a coil form 36 which surrounds the shell 16. As shown best in cross section, FIG. 2, the first windings are substantially equally spaced about the minor circumference of the coil form 36, which may be in the form of two halves bolted together as illustrated. A plurality of second windings 38 are wound substantially helically upon the coils form substantially midway between respective successive first windings. Each of the windings 34 and 38 may be formed of a plurality of turns of conductors 40 which may be square in cross section and insulated from one another. The conductors 40 may include central passages 42 for the circulation of coolant for cooling the conductors. The first and second windings 34 and 38 are regarded as helical even though they do not form true helices in the sense of being wound upon circular cylinders. The windings 34 and 38 are wound uniformly as they progress around the torus so that the first windings upon making a complete circuit of the torus register with first windings so as to form continuous first windings all the way around the major axis of the torus. That is, where there are two first windings, the number of turns must be integral or half way between. In the latter case, what was one first winding the first time around is the other first winding the second time around. The same thing is true for the second windings 38. The first windings 34 are energized by a direct current source 44, and the second windings 38 are energized by a direct current source 46. The direct current sources 44 and 46 are oppositely poled so as to pass current through the respective first and second windings in opposite directions. Such currents provide a steady state helical magnetic field within the primary chamber 14 for combining with the poloidal magnetic flux produced by the plasma current for the purpose of containing the plasma current away from the conductive walls of the liner 12. The helical windings 34 and 38 are preferably wound at such pitch as to produce relatively small interwinding forces and good plasma stability. An angle of about 45.degree. to the minor axis of the torus is suitable. As shown in FIGS. 1, 2, and 3, there may be two first windings and two second windings disposed about the minor circumference of the torus. Three of each such windings can also be used, filling the primary chamber more fully with plasma but possibly with less stability. A greater number is possible under some conditions. The power supplies are connected so that the current through the first windings can be equal to or slightly greater than the current through the second windings, whereby a zero or a net toroidal magnetic field is produced by the helical windings 34 and 38. In general, the total current in the second windings 38 is comparable in magnitude to one half the plasma current. The additional turns 32 can be operated to apply a vertical magnetic field to the plasma as to balance the effect of hoop force which tends to expand the plasma in major radius, or to adjust the equilibrium plasma for best stability. The device may include observation ports 48. In typical operation of a device as shown in FIGS. 1 through 5, the plasma current generated by operation of the solenoid coil 30 and additional turns 32 is about 40 kA maximum, which requires a magnetic flux swing of about 0.3 V-sec. with a rise time of about 10 msec. To achieve a ratio .beta. of plasma pressure to magnetic field pressure of about 0.1 while maintaining good stability, typically the temperature T of the plasma will be about 100 eV, with a density n of 10.sup.13 particles per cc., a magnetic flux density B of 1 kilogauss, an energy containment time .tau..sub.E of 0.3 msec. and a pulse duration .tau..sub.pulse of 30 msec. The total current in the first windings is about 20 kA and in the second windings is also about 20 kA. The ratio of the mean radius of the plasma current r.sub.p to the mean radius of the windings r.sub.w is about 0.75. Under such conditions, the equilibrium profiles of certain parameters have been calculated to be qualitatively as shown in FIG. 6. The relationships among the various parameters of the system and their relationships to the operation of the system are complicated and depend upon many different factors. For the sake of explanation, the curves of FIG. 6 have been prepared based upon certain parameters which have been selected somewhat arbitrarily. For the curves illustrated, the aspect ratio of the primary chamber 14, that is, the ratio of major to minor radii of the torus, is high. More particularly, the parameters there illustrated are: j.sub.z, the current density in the direction of the minor axis of the torus; j.sub..theta., the current density in the direction around the minor axis; B.sub.z, the net magnetic flux density in the direction of the minor axis; B.sub..theta., the magnetic flux density around the minor axis, and q, the safety factor related to B.sub.z and the pitch of the magnetic field lines as defined previously. The parameter r/r.sub.s is the ratio of the minor radius coordinate to the minor radius of the separatrix, this ratio evaluated along an angle of 45.degree. to the X/r.sub.s axis of FIG. 7. FIG. 7 illustrates the magnetic flux surfaces generated under these conditions at points A and B of FIG. 6. A condition for stability is that q pass through zero. The manner of operation of this invention with the resultant stable plasma can be described mathematically. The mathematics, however, becomes very complex for certain configurations. If certain practical approximations are made, the explanation can be much simplified. For example, as a practical matter it is desirable to operate with a high aspect ratio; that is, the ratio of the major radius to the minor radius of the torus can be very large, somewhat like a bicycle tire. In such cases the toroidal effects can be neglected in favor of a cylindrical approximation. The main field is B.sub..theta.,o (r) produced by the plasma current. A helical winding produces a magnetic field given by a static potential .PHI., EQU .PHI.=(b/k)I.sub.l (kr) cos (l.theta.+kz) (3) where I.sub.l is the modified Bessel function of order l. The components of the magnetic field are given by ##EQU5## Here I.sub.l ' (kr) is the derivative of I.sub.l (kr) with respect to its argument. The entire field may be expressed in terms of the flux function .psi.* given by EQU .psi.*=.psi..sub.o *-(br/l)I.sub.l '(kr) sin (l.theta.+kz) (7) where .psi..sub.o *=-.intg.B.sub..theta.,o dr. Surfaces defined by .psi.*=const. are the flux surfaces. The shapes of the flux surfaces may be calculated approximately by setting ##EQU6## By expansion, ##EQU7## It follows EQU .xi..about.-{(ba/l)I .sub.l '(ka)/(B.sub..theta.,o)} sin (l.theta.+kz). (11) The translational transform may be calculated by the flux line equations ##EQU8## By using expansion (8) ##EQU9## The average value is then given by ##EQU10## The safety factor q is as defined above ##EQU11## The volume .DELTA.V between two flux surfaces .psi.* and .psi.*+.DELTA..psi.* may be calculated from ##EQU12## By using r=a+.xi., ##EQU13## The longitudinal flux .psi. is calculated from EQU .psi.=.intg.B.sub.z rdrd.theta. (17) By using Eqs. (6), (8), and (10), ##EQU14## The combination of Eqs. (16) and (18) yields ##EQU15## It is a decreasing function of a and indicates d.sup.2 V/d.psi..sup.2 &lt;0. This characteristic has been called a magnetic well; C. Mercier, "Lectures in Plasma Physics", Fontenay-aux-Roses (1974). In the limit of .beta..fwdarw.0, this assures stability. More intuitively, the magnetic well means that the average longitudinal magnetic field increases as one moves away from the plasma, where "average" means flux-surface average. The maximum well depth occurs for r=0. Using Mercier's notation, the magnetohydrodynamic equilibria in the cylindrical approximation may be calculated using the equation given by ##EQU16## The helical variable u=l.theta.-hz in the cylindrical coordinates (r,.theta.,z) and the vector u=(le.sub.z +hre.sub..theta.) (l.sup.2 +h.sup.2 r.sup.2).sup.-1 define the helix. The magnetic field B is written as EQU B=fu+ux grad F (21) The operation L is defined by ##EQU17## It is convenient to use the variable G defined by ##EQU18## Then ##EQU19## It is instructive to calculate a simple example equilibrium, where f=const. and p'=const. Then Eq. (20) becomes ##EQU20## By integrating ##EQU21## where C is a constant. Putting G=G.sub.o (r)+g(r,u) (27) ##EQU22## This is a special case where the vacuum field g is separated out. In order to avoid the singularity on the axis in the absence of the internal conductor, ##EQU23## By integration of Eq. (28), ##EQU24## The external vacuum field g is given by EQU g=(b/h)I.sub.l (hr) sin (l.theta.-hz) (32) The function F is then given by ##EQU25## The magnetic fields are ##EQU26## If there is no solenoidal field applied, then the axial field vanishes at the plasma edge r=r.sub.o. Then EQU f/l=-(hr.sub.o.sup.2 /2)p' (36) This indicates that the plasma produces a paramagnetic axial field of f/l on the axis. On the other hand, if f=0, an external field of -(hr.sub.o.sup.2 /2) p' is required. The plasma is diamagnetic to this field. The current density j is given by ##EQU27## The azimuthal component is ##EQU28## Obviously f=const. does not lead to small j.sub..theta.. The equilibria with small j.sub..theta. are the ones of interest. Consider a case where f=(2h/l)F and p'=const. The equilibrium equation is given by ##EQU29## Putting EQU F=-(l.sup.2 /4)p'r.sup.2 +H (40) Eq. (39) becomes ##EQU30## In this case, the pressure is supported by azimuthally symmetric j.sub.z B.sub..theta. force and the helical field H is a force-free field. The field and the current are given by ##EQU31## Note that only j.sub.z and B.sub..theta. have non-helical components. Equations 42-47 describe an equilibrium which has no non-helical contribution to B.sub.z on axis. On the other hand, the equilibrium described by equations 34-38 has a very large non-helical B.sub.z component. In between these two equilibria lie equilibria that have an intermediate B.sub.z component to give an appropriate q profile. Thus, by superposing the two example equilibria described, an equilibrium of a desired amount of the solenoidal axial field may be obtained. FIGS. 6 and 7 illustrate qualitatively the type of equilibrium which is desired. Such an equilibrium is expected to be stable according to Mercier's criterion for beta values in excess of 10%. Mercier's criterion, which must be satisfied for the plasma to be stable, is given by ##EQU32## The quantity .XI. as used by Mercier is proportional to the pressure gradient and the last term corresponds to the destabilizing effect of the pressure. The criterion reduces to the Suydam's criterion for a cylindrical pinch given by ##EQU33## It has been known that pinches can be made stable by profiling B.sub.z and q. The outer part of the plasma is stabilized by a large shear and a small .beta. with respect to the axial field. The inner part is made stable by having a hollow pressure distribution. In these configurations, the axial field is reversed, i.e., there is a null of the axial field in the plasma. The profile must be maintained for the stability throughout the duration of the discharge. This is one of the experimental difficulties of the reversed field pinch. If B.sub.z is taken to represent the axial transform in the criterion, the outer part of the plasma is stabilized because of the shear and a large transform. The inner part has to be stabilized by an axial field produced by the plasma current and/or by unbalancing the current in the helical windings to counter the axial transform, thus having a q profile similar to the reversed field pinch. At any rate, the q profile in this case is externally controlled. The amount of the axial field is controlled by unbalancing the current in the positive and the negative helical windings. A proper q profile can be maintained independent of the plasma skin time. Relating this physically to the structure illustrated in FIGS. 1-5 and to the curves of FIGS. 6 and 7, the twisted magnetic field produced by plasma current and the helical magnetic field produced by windings 34 and 38 result in magnetic flux surfaces wherein the safety factor q as a function of radial displacement from the minor axis of the toroid has a substantial slope and changes monotonically, reversing sign near the outer edge of the plasma. By adding or subtracting a small amount of toroidal magnetic flux relatively uniformly across the torus, the net toroidal flux as a function of radial displacement can be moved up or down to cross zero at an optimum radius for confining the plasma. Such additional toroidal magnetic flux is generated by the unbalance of the helical magnetic fields produced by the respective first and second windings 34 and 38. As defined above, a flux surface is a surface on which the magnetic flux density, evaluated at any point on the surface, has no component normal to the surface. In other words, a flux surface is a surface which no magnetic field lines penetrate. The field lines lie on the flux surfaces. The flux surfaces are nested. A criterion for stability is that the flux surfaces be nested and separated from the confinement wall. In a toroidal configuration, the flux surfaces must be closed. Thus, in accordance with the present invention, the combination of the poloidal magnetic field produced by the plasma current and the helical magnetic field produced by the helical windings provide a magnetic limiter separating the plasma current from the confinement wall of the plasma vessel. This creates the separatrix, which defines a closed surface which limits and encloses the region within which the closed and nested flux surfaces exist. As defined above in equation 1, ##EQU34## where q is the safety factor, R is the major radius of the torus, and dz/d.theta. is the average length traversed in the toroidal direction per unit poloidal angle of rotation of a magnetic field line on a magnetic flux surface. In accordance with this definition, an average magnetic field line in a flux surface makes q transits around the torus in the toroidal direction in making a single transit in the poloidal direction. (In the present case, q is a fraction which is less than 1.) Thus, the safety factor q on a particular flux surface is the ratio of the average pitch of magnetic field lines in that flux surface to the major circumference of the torus, where pitch is the displacement in the toroidal direction for a single transit, or cycle, in the poloidal direction. As stated by equation 12, dz/d.theta. is also given by ##EQU35## where r is the minor radius, B.sub.z is the longitudinal or toroidal magnetic field and B.sub..theta. is the poloidal magnetic field. ##EQU36## is the translational transform. Thus, ##EQU37## where the angular brackets indicate an average over a flux surface. For circular concentric flux surfaces in an axisymmetric system, the average is a simple average over the poloidal angle .theta.; that is, ##EQU38## but since neither B.sub.z nor B.sub..theta. depends strongly on .theta., ##EQU39## for such case. Equation 52 is appropriate for a tokamak or a reversed field pinch. For tokamaks, q is greater than 1 everywhere, and for the reversed field pinch, q vanishes only when B.sub.z vanishes. In such case, B.sub.z is a net toroidal field, meaning that it persists when averaged over poloidal angle .theta.. In the case of the present invention, in the embodiment where the currents in the helical windings are balanced, there is no net B.sub.z except that due to poloidal plasma currents. However, there can be an average B.sub.z on a flux surface. This may be understood by reference to FIG. 8, which is a simplified version of FIG. 2. The windings 34 and 38 are represented by single conductors and the rest of the apparatus is omitted for the sake of clarity in this explanation. Dashed lines 50 and 52 have been drawn to separate the space in the chamber 14 into quadrants. On these lines, the toroidal magnetic field is zero. In quadrants I and III, the toroidal field is caused by the first windings 34 and is directed up out of the plane of FIG. 8 for the twist as shown. In quadrants II and IV the toroidal field is opposite to this. The toroidal field averaged over a circular loop 54 is zero, because it passes equally through all four quadrants. If the circle is distorted into an ellipse 56, the toroidal field averaged over the loop is now non-zero. For the loop 56, the path is longer in quadrants I and III and shorter in quadrants II and IV. Also, the path is nearer to the first windings 34 in quadrants I and III, where the toroidal field is stronger, and farther from the second windings 38 in quadrants II and IV, in a reduced toroidal field. Both the extra path length and larger field weight the average to have quadrants I and III dominate. This makes an average toroidal field on the loop 56 which is directed up out of the plane of FIG. 8. Near the center of the plasma the net toroidal field is generated by poloidal plasma current. At a point near the edge of the plasma the effect of the remaining poloidal plasma current, that which remains between that point and the edge, is relatively much smaller and can be overcome by the flux-surface-average toroidal field due to the helical coils. This gives the q reversal with balanced coils when the appropriate currents and fields are applied with proper polarity. The device of the present invention as thus described differs fundamentally in both principle and structure from the prior art devices as exemplified by tokamaks, stellarators and reversed field pinch devices, although the present device has certain features in common with each. More particularly, like the tokamak, the present device requires plasma current to generate the appropriate magnetic flux configuration, and the configuration does not decay on the flux diffusion time scale. On the other hand, the tokamak requires toroidal field coils and not helical field coils; whereas the present device requires helical field coils but not toroidal field coils. The tokamak requires q greater than 1; whereas the present device does not. The present device requires q to cross zero as a function of radial displacement; whereas the tokamak does not. Like the stellarator, the present device requires helical field coils; but unlike the stellarator, it does not require toroidal field coils. As in the stellarator, the magnetic configuration does not decay on the flux diffusion time scale, but unlike the present device, the stellarator does not require plasma current to generate the magnetic configuration. The stellarator requires a large toroidal flux B.sub.z ; whereas the present device does not require any net applied toroidal flux, although a small applied B.sub.z may be desirable for optimization. The present device requires that q cross zero as a function of radial displacement, which the stellarator does not. In contrast, a stellarator with a substantial plasma current generally requires q&gt;1 for stability. Like reversed field pinch devices, the present device requires plasma current to generate the magnetic configuration and for q to cross zero. Neither requires q greater than 1. On the other hand, the present device requires helical coils, which the reversed field pinch devices do not, and has a separatrix, which the reversed field pinch does not. The magnetic configuration decays on the flux diffusion time scale in reversed field devices but not in the present device. These differences and others provide substantial advantages for the present device. The fact that no large toroidal magnetic field is required permits great economy in manufacture and ease of operation as makes this a more practical device. The large toroidal magnetic fields required for tokamaks and stellarators apparently require superconducting magnetic coils and imply large interwinding forces that produce stresses difficult to contain. Prior devices have often involved neutral beam heating which has proven inefficient, bulky, and expensive, and has caused problems when the beam hit a wall. The present device in general is relatively smaller, being capable of high beta, high aspect ratio, and no applied net toroidal field, and permits adequate ohmic heating in conjunction with the induced plasma current. A problem, particularly with tokamaks, has been the relatively small space available for the plasma heating coils. The present device in its preferred form has a relatively large aspect ratio, permitting more space for such coils and other appurtenances such as a reactor blanket. This eases the design requirements of the heating coils. This also permits scaling to larger devices merely by increasing the major radius while keeping the minor radius the same. The present design provides a higher .beta., the ratio of plasma pressure to magnetic pressure, permitting more efficient operation at lower magnetic fields. The present design provides an inherent magnetic limiter whereby the separatrix moves radially outward as the plasma current increases, maintaining a stable configuration. This is because outside the confining flux surface there it no confinement and any plasma outside the separatrix is immediately lost to the confining wall without wastefully carrying any substantial current. The present design also facilitates the incorporation of a divertor, which is difficult to introduce in tokamaks. An advantage over the reversed field pinch devices is that such devices operate with a q profile that changes for the worse as the magnetic flux diffuses out of the system. The time is so short as to have severely limited the development of a practical reactor based on the reversed field concept. While the novel aspects of a fusion device in accordance with the present invention have been shown in a preferred embodiment, various modifications may be made therein within the scope of the invention, as in the size and shape and in driving currents. For example, the direct current in the windings 34 and 38 may take the form of relatively long unidirectional pulses. The device may also include various well-known appurtenances of fusion devices such as power supplies, vacuum pumps, instrumentation, blankets, supporting structures, and heat exchangers. Although the preferred embodiment of the invention is a toroidal system, the invention may also be utilized in a straight cylindrical system appropriately bounded. As the length L of a toroidal system is the major circumference 2.pi.R, the safety factor q may be defined in terms of L: ##EQU40## This safety factor as thus defined is applicable to a straight cylindrical system of length L.