europhysics BULLETIN OF THE EUROPEAN PHYSICAL SOCIETYnews J.A. Volume 17 Number 6 Directory June 1986

coil geometry, similar to a simple magne­ The Spheromak tic mirror machine, and offers conside­ rable simplification. Stephen C. Jardin, Princeton If we follow an individual magnetic field line in Fig. 1 as the toroidal angle ϕ (Plasma Physics Laboratory, Princeton University) increases around the device, we see that it remains in a two-dimensional surface Intuitively, the most attractive form for a confined thermonuclear fusion ψ(R,Z) = constant. These constant ψ plasma is a simple sphere which avoids all the problems associated with values form nested toroidal surfaces en­ an inner vessel wall. compassing the magnetic axis. Since both electron and ion orbits make tight Among the most promising, although Confinement Geometry spirals around individual magnetic field least mature, of the magnetic confine­ The spheromak is related to but dis­ lines, these nested magnetic surfaces ment geometries proposed for contain­ tinct from several other magnetic confi­ form the lowest order basis for the sphe­ ing high temperature deuterium-tritium nement concepts. Like the tokamak and romak confinement scheme, as they do fusion plasmas is the spheromak con­ reversed field pinch (RFP), the sphero­ for tokamaks and RFP's. We characte­ figuration. The simplicity and compact­ mak magnetic fields are axisymmetric, rize these as closed field line geometries, ness of this device has been the impetus or independent of the cylindrical angle ϕ, as opposed to open field line systems for innovative and novel theoretical and Fig. 1. Also, like these other devices, like magnetic mirror machines where in­ experimental investigations during the large currents flow in the plasma itself dividual magnetic field lines pass out of past decade. Although still considered which produce magnetic fields which in the confinement region and may inter­ an "alternative" or "supporting" con­ turn confine the plasma. The spheromak sect the surrounding walls. The open cept, the spheromak is now beginning to differs, however, in that only one set of field line confinement physics relies on demonstrate an ability to heat and con­ external coils are necessary to provide anisotropy in the velocity space distribu­ fine plasmas approaching those of ther­ the required externally generated ma­ tion function and on the establishment monuclear interest. gnetic fields. This leads to a non-linked of confining electrostatic fields, whereas closed field line confinement does not. It is useful to think of the spheromak axisymmetric magnetic field as being composed of two parts, a poloidal part

C ontents The Spheromak 73 EPS Directory 77 Members 77 Executive Committee 77 Advisory Committees 79 Divisions and Sections 80 Associate Members 82 Fig. 1 — The spheromak Meetings Up-date 82 equilibrium is independent of Charge Exchange in Atomic the cylindrical angle

Europhysics News is published monthly by the European Physical Society. © 1986. Reproduction rights reserved. ISSN 0531-7479 73 constrained by axisymmetry and the volume as condition V · B = 0 to be of the form K = ∫ A· BdV, (3) Bp = (2 ) -1V(p x Vψ/(R,Z), and W = ∫ 1/2 B2 dV, (4) and a toroidal part where A is the vector potential associa­ BT = g(R,Z)Vψ. ted with the magnetic field B, ie, B = V The total magnetic field B = Bp + BT is x A. Because of the gauge freedom in thus completely described by these two A, the transformation A → A + Vx two-dimensional scalar functions ψ(R,Z) leaves B unchanged for any scalar field and g(R,Z). Under certain mild assump­ X, even one that is multi-valued. To be tions, namely that the plasma is a non­ definite we must also stipulate that A is rotating equilibrium fluid adequately single valued in the sense that the line in­ characterized by a scalar pressure, it can tegral of ∫ A · dl around any closed curve easily be shown that both the toroidal lying within the volume is equal to the magnetic flux B · dS through the area field function g(R,Z) and the fluid J Fig. 2 — Helicity K is a measure of the pressure p(R,Z) must be single valued enclosed by the curve. topological linkage of the magnetic field. If functions of the poloidal flux function A simple example serves to illustrate two magnetic flux tubes with flux ϕ1 and ϕ2 ψ(R,Z), i.e., g = g(ψ) and p = p(ψ). that the magnetic helicity K measures interlink, helicity is K = 2ϕ1ϕ2. These functions are related to each the topological linkage of the magnetic as first noted by J.B. Taylor 1), in such a field. Consider the situation in Fig. 2 other by the force-balance, or equili­ plasma the changes in field topology are brium equation: where we have two intertwined magne­ accompanied by only very small chan­ J x B = Vp (1) tic flux tubes, one containing flux ϕ1 and ges in the field itself and the integral K = the other containing flux ϕ By defini­ or, using Ampere's law V x B = µQJ, 2 . ∫ A · B dr over the entire volume will be (2 )-2R2V · R 2Vψ + tion, the magnetic field is everywhere almost unchanged. The effect of the µ0R2dp/dψ + gdgldψ = 0. (2) tangential to the boundaries of the flux topological changes is merely to redistri­ The distinguishing feature of the tubes so that the magnetic flux, ϕ = J B bute the integrand among the field lines spheromak configuration is that the • dS, through any cross-section of either involved. If surrounded by perfectly con­ toroidal field, and hence g(ψ), goes to of the tubes is a constant, equal to ϕ1 or ducting walls, the global helicity K will zero at the last closed surface separa­ ϕ2 . Assuming B to vanish in the volume still be a good invariant even though the ting the confined plasma region from the outside the two flux tubes, we can easily individual helicity on each flux tube is surrounding vacuum region. This im­ calculate the helicity by integrating over not. plies, through Ampere's law, that no the volumes inside, Taylor's hypothesis, then, is that a toroidal field magnets are needed to pro­ K = ∑ ∫ A · B dV = non-perfect plasma will undergo some duce the spheromak toroidal fields, as is ∑ ∫ A · dl B · dS = 2ϕ1ϕ2 (5) turbulent relaxation, but it will be such the case in tokamaks and RFP's. The where the integration over each of the as to take the plasma into its lowest toroidal field is non-zero only in the ac­ two flux tube volumes gives the same possible energy state consistent with tual plasma region and is produced en­ result. We see that if the tubes were not the constraint that the global helicity K tirely by poloidal currents flowing in the linked, the helicity of the configuration does not change. One is able to show plasma itself. This feature means that it would be zero since the integral ∫ A · dl that this state is such that the electrical is not necessary for any external struc­ would be zero in each flux tube integral. current is everywhere proportional to ture to pass through the centre hole of If the spheromak plasma is modelled the magnetic field. the doughnut-like toroidal surfaces, and as a perfectly conducting fluid, without V x B = AB (6) thus the confinement region becomes electrical resistance, and surrounded by where A is a single proportionality cons­ topologically spherical as opposed to a perfectly conducting wall, it can easily tant determined by the global helicity. toroidal. The inherent simplification this be shown that the global magnetic Such an equilibrium configuration with affords is substantial. The aspect ratio of helicity K is an exact invariant. In fact, for constant A has become known as the the plasma-vacuum interface can be such a plasma, the individual helicity "Taylor State". made to approach unity. The toroidal associated with each of the infinite num­ To the extent that the Taylor principle field magnets, normally the most expen­ ber of flux tubes in the plasma is an inva­ applies, the magnetic helicity K is the sive component in a toroidal confine­ riant, remaining constant in time as the fundamental property of the sphero­ ment device, are eliminated. Shielding plasma deforms or moves. This state­ mak. Global helicity is created or in­ requirements are greatly eased by the ment is equivalent to saying that in a jected during the formation scheme, and change in topology. In addition, there is plasma fluid with infinite conductivity, eventually decays to zero due to resis­ the possibility that the spheromak struc­ magnetic field lines are frozen into the tive dissipation. At any intermediate ture can be translated in the direction of fluid, and since the fluid velocity is conti­ time the fields and currents are uniquely the symmetry axis, raising the possibili­ nuous, magnetic field lines cannot break determined by Eq. (6) with A being deter­ ty of separate formation, thermonuclear or coalesce to change their topological mined by the instantaneous value of the burn, and disposal regions. properties. global helicity. Magnetic Helicity The special significance of the global The Taylor state equilibrium described A concept central to the discussion of magnetic helicity K comes from con­ by Eq. (6) is a particular solution of the the stability, formation, and decay of the sideration of a more realistic plasma more general equilibrium Eqs. (1) and (2) spheromak is that of magnetic helicity. If model in which we allow small depar­ that has g proportional to ψ and p = 0 we temporarily restrict consideration to tures from the perfect-conductivity ap­ everywhere. This state is force free, a volume with no magnetic field lines proximation. In such a plasma, topologi­ having Vp = 0 everywhere, and is thus penetrating the bounding surfaces, then cal properties of the magnetic field are unsatisfactory as a fusion equilibrium we can define the global magnetic heli­ no longer strictly preserved, and lines of magnetic confinement configuration. city and the magnetic energy in that force may break and coalesce. However, Although Eq. (6) is useful as a first ap- 74 proximation, detailed stability analysis Each of these fundamental modes of must be performed to determine what instability has a simple physical inter­ other stable equilibria are present in the pretation. The tilting mode arises since neighbourhood of the Taylor state that the magnetic moment of the spheromak allow a finite pressure gradient and thus is aligned antiparallel to the external ver­ provide confinement. Also, in a geome­ tical field. A lower energy state can be try without conducting walls in contact obtained by the plasma tilting 180°, with the plasma, the Taylor principle cer­ although it will no longer be in radial tainly does not apply and mode analysis force balance when it does so. The shif­ must be performed to locate instabilities ting and vertical modes correspond to and stable configurations. the plasma ring displacing itself into a region of weaker external field strength, Stability and thus lowering its interaction energy. The central question one asks of any Although several innovative methods proposed magnetic confinement sche­ for controlling these instabilities with me is its stability. The spheromak con­ energetic particles or plasma rotation figuration is assumed to be symmetric have been proposed, the only method with respect to rotations in the cylindri­ which has been experimentally demons­ cal angle 0 (axisymmetric), and to pos­ trated relies on the placement of exter­ sess symmetry with respect to reflec­ nal conductors in close proximity to the tion about the midplane. We must ask shell. Induced currents appear in these whether any perturbation allowed by the conductors when the plasma is displa­ dynamics which breaks this symmetry ced, producing a restoring force which is energetically favourable, that is, will tends to push the spheromak back to its take the configuration to a lower energy symmetric state. Although effective, the state. If so, the system is unstable. necessity for having solid conductors in There are many levels of mathemati­ close proximity to the plasma compro­ cal description for a magnetically con­ mises its flexibility and desirability. Also, fined plasma. These differ in their com­ active feedback systems are necessary plexity and sophistication, and in their to stabilize these motions for times long realism in describing the true physical compared to the resistive decay times of situation. It is useful to classify instabili­ the conductors. ties with respect to what level of des­ The next useful level of description is cription is necessary for the instability to the "ideal MHD" model 3) where we appear. This classification also clarifies allow non-rigid deformations of the plas­ the free energy driving the instability ma, but treat it as a perfectly conducting and the magnitude and scaling of the fluid. Extensive analysis of the stability growth rates for unstable perturbations. of the spheromak using the ideal MHD Perhaps the most troublesome of the model shows new instabilities if the Fig. 3 — Rigid instabilities include (a) tilting, spheromak instabilities are the tilting, plasma pressure is too high (pressure (b) shifting, and (c) vertical motion. the shifting, and the vertical mode 2). driven) or if the toroidal current channel These are global, rigid displacements of is too peaked (current driven). These in­ current flowing parallel to the magnetic the plasma, present in its simplest des­ stabilities, while also having large field lines, in which case we call them cription, that of a rigid current carrying growth rates comparable to the rigid "tearing" modes. ring. A spheromak carrying a toroidal mode growth rates, are avoidable if we The resistive modes are predominant­ current /p in the positive 0 direction re­ limit the maximum pressure and current ly localized in small bands around "ra­ quires an externally supplied magnetic density. These ideal MHD stability limits tional" magnetic surfaces in which the field with a component in the negative z- are given approximately 5,6) by: µ0p0 < magnetic field lines close upon them­ direction, to produce a radially inward B2o/10, and µ0lp < 10 aB0 (mks units) selves after traversing m times the short force which balances its intrinsic radially where p0 and Bo are the central values of way around the torus and n times the outward directed expansion force. This the plasma pressure and toroidal field, lp long way around. An unstable resistive external field can be adequately charac­ is the total plasma toroidal current, and a mode will develop large perturbed cur­ terized as having a strength Bz0 and a is the plasma minor radius. These limits rents in the resistive inner region sur­ magnetic field index n = - (r/Bz0)(∂Bz0/∂r). are not unduly restrictive. rounding one of these rational surfaces It is easily shown that the rigid sphero­ If one next adds finite electrical resisti­ causing increased magnetic reconnec­ mak ring will be unstable to at least one vity to the fluid model of the plasma, the tion and transport of plasma across the of the modes of Fig. 3 for any value of stability picture changes considerably. magnetic field lines. Rational magnetic the field index n. If n < 0 the vertical Normal mode analysis says that a new surfaces with m = 1 and n = (small in­ mode is unstable, if n > 0 the shifting class of resistive instabilities are pre­ tegers) have the most stringent stability mode is unstable, and if n < 1 the tilting sent. These resistive instabilities may criteria. mode is unstable. The growth rates for tap predominantly the free energy The stability criteria with respect to these instabilities are large, scaling like source associated with the plasma pres­ these resistive instabilities also limit the Y ~ (lpBz0/M) 1/2, where M is the total sure, in which case we call them "resis­ allowable current and pressure distribu­ mass of plasma. These correspond to tive interchange" or "resistive balloon­ tions in the plasma, but the limits are times of the order of 1-10 µs for present ing" modes, or they may tap the free much more restrictive. Only a small spheromak experiments. energy associated with the electrical range of current distributions are found 75 free Taylor state. Ultimately, this ques­ gnetic fields of 0.5 T. Discharge duration tion will have to be answered experi­ times up to 2 ms have been obtained on mentally. several of these devices, which have major radii in the range of 9-60 cm. Formation Methods Magnetic probe measurements con­ There are presently at least four dis­ firm that the plasma in these devices in­ tinct methods of forming spheromaks deed tends to relax into a state that lies that have been demonstrated. These near the Taylor minimum energy state. are : Specific resistive modes have been iden­ 1) by a magnetized coaxial gun using tified as effecting this relaxation, in good currents through electrons 4) agreement with the theoretical onset 2) by a combination of 6 and z pinch conditions. This implies that the final discharges utilizing both electrode and plasma state is relatively insensitive to inductive techniques 5) the details of the formation process, but 3) by a conical theta pinch technique depends primarily on the amount of ma­ utilizing fast inductive discharges and gnetic helicity injected during formation. 4) by an electrodeless inductive scheme A noteworthy experiment on the CTX utilizing a flux core 6). device has demonstrated near steady- These schemes differ in detail, but all state maintenance for over 5 ms by con­ have the common feature of producing tinuous injection of helicity from an ex­ both toroidal and poloidal field com­ ternal electrode source 4). ponents, and in affecting a change in It has been demonstrated that the glo­ topology to break off field lines that are bal tilt and shift modes can be stabilized open or linked with external structures by passive conductors. This has been to produce a free spheromak with clos­ accomplished by solid conductor walls, ed magnetic surfaces. We illustrate in by "bird cage" wire mesh enclosures, Fig. 4 the inductive flux core method and by a figure-8 coil, which is a special­ used in the S-1 Spheromak at Princeton ly designed wire loop that is twisted to University. give the current pattern needed to The S-1 flux core is a thin walled stabilize these instabilities, but to have stainless steel toroidal tube that con­ near zero inductance with the axisym- tains current-carrying windings that go metric coil systems. around both the short way (TF windings) Although the temperature and confi­ Fig. 4 — The S-1 inductive formation and the long way (PF windings). Each of nement parameters obtained to date in method begins with (a) the plasma linked to these sets of windings are connected the spheromak fall considerably short of the flux core. As currents are induced into through leads to separate external vol­ those obtained in tokamaks, the intrinsic the plasma (b) reconnexion occurs and (c) tage sources, the TF source and the PF attractiveness of this configuration bec­ the spheromak separates from the struc­ source. As these circuits are pulsed, kons us to continue its investigation. ture. large currents appear in the windings Steady improvement in these parame­ and, through inductive coupling, in the to be stable to tearing instabilities, and ters over the last few years encourages these are ones close to the Taylor state, surrounding plasma. Initially, the plasma us to believe that continuing success in Eq. (6). Also the resistive interchange or surrounds, or is linked to, the flux core the development and optimization of ballooning modes set rather low limits (4a). As the vertical field and currents this concept will be forthcoming. on the allowable pressure. The precise are increased, the plasma separates from the flux core with each magnetic pressure limits set by these instabilities REFERENCES depends on many factors, including surface reconnecting on the small major radius side (4b). When the reconnection 1. Taylor J.B., Relaxation of Toroidal Plasma details of the geometry, the collisionality and Generation of Reverse Magnetic Fields, regime, and the Larmor radius size of the is complete, a free non-linked sphero­ Phys. Rev. Lett. 33 (1974) 1139. ions. However, it is clear that they set mak plasma has been created (4c). 2. Rosenbluth M.N. and Bussac M.N., MHD limits as much as an order of magnitude Stability of Spheromak, Nucl. Fusion 19 less than the pressure limits set by ideal Experimental Results and Prospects (1979) 489. modes. Over half a dozen spheromak devices 3. Jardin S.C., Ideal Magnetohydrodynamic Stability of the Spheromak Configuration, The growth rates associated with the are now operating in the USA, Japan, Nucl. Fusion 22 (1982) 629. resistive modes are several orders of and Europe. Experiments exist in the 4. Jarboe T.R., Henins I., Sherwood A.R., magnitude smaller than those for ideal USA at the University of Maryland, Barnes Cris W. and Hoida H.W., Slow Forma­ modes. This, coupled with the highly LANL, PPPL, and the University of tion and Sustainment of Spheromaks by a localized nature of these instabilities Washington; in Europe at the Univer­ Coaxial Magnetized Plasma Source, Phys. makes it plausible that their effects are sities of Essen and of Heidelberg, and in Rev. Lett. 51 (1983) 39. rather benign, serving only as a mecha­ Japan at the Universities of Osaka and of 5. Goldenbaum G.C., Irby J.H., Chong Y.P. nism for allowing the plasma to "relax" Tokyo. Temperatures of the order of 100 and Hart G.W., Formation of a Spheromak back to a configuration near the Taylor ev, sufficiently high to overcome the Plasma Configuration, Phys. Rev. Lett. 44 (1980) 393. state once it has deviated too far. The ef­ low-z impurity radiation barrier, have 6. Yamada M., Furth H.P., Hsu W., Janos A., ficiency and desirability of the sphero­ been achieved on several of these devi­ Jardin S., Okabayashi M., Sinnis J., Stix T.H. mak as a fusion plasma confinement ces. Densities are typically 0.3 to 10 x and Yamazaki K., Quasistatic Formation of configuration depends on how far on the 1014 cm -3, with toroidal currents in the the Spheromak Plasma Configuration, Phys. average it can deviate from the force- range 0.1 to 1.0 MA, producing peak ma­ Rev. Lett. 46 (1981) 188. 76