∫° Plasma Confinement in a Levitated Magnetic Dipole

MAGNETIC CONFINEMENT SYSTEMS

Plasma Confinement in a Levitated Magnetic Dipole J. Kesner* and M. Mauel** * Massachusetts Institute of Technology, Cambridge, Ma 02129 ** Columbia University, New York, N.Y. 10027 Received April 1, 1995

Abstract—Plasma confinement in the field of a levitated dipole offers many advantages for magnetic fusion. MHD stability is obtained from compressibility which utilizes the large flux tube expansion of a dipole field. Such a device could be high beta, steady state, and exhibit good confinement properties. The large flux expansion will ease the difficulty of the divertor design. The configuration is ideal for electron cyclotron heating and controlled convective flow patterns may provide a mechanism for fueling and ash removal.

1. INTRODUCTION tribution function, µ is the adiabatic invariant, µ = The dipole magnetic field is the simplest and most ⑀⊥/2B, J is the parallel invariant, J = °∫v || dl and the flux, common magnetic field configuration in the universe. It ψ is the third adiabatic invariant. The frequencies that is the magnetic far-field of a single, circular current Ω correspond to these invariants are respectively c the loop, and it represents the dominate structure of the cyclotron frequency, ω , the bounce frequency and ω middle magnetospheres of magnetized planets and neu- b d tron stars. The use of a dipole magnetic field generated the curvature driven precessional drift frequency. Both by a levitated ring to confine a hot plasma for fusion of these conditions lead to dipole pressure profiles that scale with radius as r–20/3 while the adiabatic distribu- power generation was first considered by Akira Haseg- µ awa after participating in the Voyager 2 encounter with tion function, F( , J) also implies a density dependence ∝ –4 ∝ –8/3 Uranus [1]. Hasegawa recognized that the inward diffu- of ne r and a temperature dependence T r . sion and adiabatic heating that accompanied strong By levitating the dipole magnet in order to prevent magnetic and electric fluctuations in planetary mag- end losses, conceptual reactor studies supported the netospheres represented a fundamental property of possibility of a dipole fusion reactor [3, 4]. The dipole strongly magnetized plasmas not yet observed in labo- reactor concept is a radical departure from the better ratory fusion experiments. For example, it is well- known toroidal-based magnetic fusion reactor con- known that global fluctuations excited in laboratory cepts. For example, the most difficult problems for a fusion plasmas result in rapid plasma and energy loss. tokamak reactor are the divertor heat dissipation, dis- In contrast, large-scale fluctuations induced by sudden ruptions, steady state operation, and an inherently low compressions of the geomagnetic cavity (due to beta limit. Furthermore, the tokamak is subject to neo- enhancements in solar wind pressure) or by unsteady classical effects and micro-turbulence driven transport. convections occurring during magnetic substorms The dipole concept provides a approach to fusion energize and populate the energetic electrons trapped in which solves these problems. the Earth’s magnetosphere [2]. The fluctuations induce inward particle diffusion from the magnetospheric 1. Divertor problem: The difficulty in spreading the boundary even when the central plasma density greatly heat load at the divertor plate is generic to concepts in exceeds the density at the edge. Hasegawa postulated which the magnetic flux is trapped within the coil sys- that if a hot plasma having pressure profiles similar to tem. By having the plasma outside of the confining coil those observed in nature could be confined by a labora- the magnetic flux can be sufficiently expanded to sub- tory dipole magnetic field, this plasma might also be stantially reduce divertor heat loads. immune to anomalous (outward) transport of plasma energy and particles. 2. Major disruptions: A tokamak has a large amount of energy stored in the plasma current. The dipole The dipole reactor concept is based on the idea of plasma carries only diamagnetic current and is inher- generating pressure profiles near marginal stability for ently free of disruptions. Experiments observe relax- low-frequency magnetic and electrostatic fluctuations. ation oscillations, reminiscent of tokamak ELMs, From ideal MHD, marginal stability results when the which relax the pressure gradients when they become pressure profile, p satisfies the adiabaticity condition, too large. δ(pVγ) = 0, where V is the flux tube volume and γ = 5/3. From gyro kinetics, marginal stability results when 3. Steady state: A tokamak is a pulsed device and ∂F(µ, J, ψ)/∂ψ = 0 where F(µ, J, ψ) is the particle dis- current drive schemes that are required for steady oper-

PLASMA CONFINEMENT IN A LEVITATED MAGNETIC DIPOLE 743 ation appear to be costly. The dipole plasma is inher- a substantially improved approach to magnetic fusion ently steady state. energy production. 4. Beta limit: Tokamak stability depends on the poloidal field which is less than the toroidal field by 2. DIPOLE CONFIGURATION Bp/BT ~ 1/qA with q the edge safety factor and A the aspect ratio. For a dipole there is a critical pressure gra- Figure 1 which shows the magnetic flux contours dient that can be supported (due to the compressibility and the mod-B surfaces for a laboratory-size levitated term in the interchange stability criterion) and for a suf- dipole facility [11]. The levitated ring has a major ficiently gentle pressure gradient the dipole plasma radius Rc and a minor radius a. The simplest vacuum resides in an absolute energy well and is stable up to chamber geometry would be a spherical tank and the local beta values in excess of unity. last closed flux surface will be assumed to have a radius ≥ 5. Transport and neoclassical effects: The trapping Rw. Typically Rw 5Rc to permit a substantial reduction of particles in regions of bad curvature makes the toka- of plasma pressure. The plasma pressure will attain a mak susceptible to drift frequency range trapped parti- peak value at a midplane radius of R0. Between the ring cle driven turbulent transport. A dipole could, theoreti- surface (located at R = Rc + a) and the peak pressure cally, show classical transport. In addition a tokamak location there is good curvature and since the field is has a “neoclassical” degradation of transport that relatively high near the ring, cross-field transport is be derives from the drifts of particles off of the flux sur- expected to be low. Thus we expect high pressure gra- faces. In a dipole the drifts are toroidal and they define dients in this region. The schematic midplane pressure the flux surfaces. profile is shown in Fig. 2. The chief drawback of the dipole approach is the need for a levitated superconducting ring internal to the plasma. Although this provides a challenge to the engi- Z, Òm neering of the device recent advances in high tempera- –300 ture superconductors coupled with an innovative design concept of Dawson [5, 6] on the maintenance of an 5 × 10 internal superconducting ring in the vicinity of a fusion –3 plasma leads to the conclusion that this issue is techno- 1 × logically solvable. –200 10 –2 Utilization of electron cyclotron resonance heating (ECRH) and pellet injection in a levitated dipole per- mits a unique heating approach for the creation of fusion grade plasmas. The combination of closed field –100 lines and high beta makes the dipole an ideal geometry for creating a high beta, hot electron plasma by the application of ECRH. Such plasmas have been demon-

strated in open ﬁeld line mirror machines as well as in 2

× levitrons and in the non-levitated dipole (or so-called 0 10

“terrella”) CTX experiment [7Ð10]. The density of the –1

5 5

10 hot electron plasma is observed to be limited by micro- ×

wave accessibility and the energy by relativistic detun- –1

1 1

× ing. Unlike ECRH mirror applications, the hot elec- 10 trons created in a levitated dipole can only be lost by 100 –1 cross-field transport and if transport is close to classical the resulting loss rate will be small. The energy stored in the hot electron plasma can be transferred into a dense, thermal hydrogenic plasma by injecting pellets into the ECR heated plasma. 200 We conclude that the continued advances in the understanding of magnetospheric plasmas coupled with the development of the technology of gyrotrons and of pellet injectors provides the fusion community 300 with a new and unique approach to creating fusion 0 100 200 300 400 grade laboratory plasmas. In addition the development R, Òm of high temperature superconductors and of new conceptual approaches to the cooling of an internal levi- Fig. 1. Magnetic field and flux surfaces for a levitated dipole tated superconducting coil provide a potential path for experiment.

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997

744 KESNER, MAUEL

p(R) The vacuum field of a circular loop provides a useful analytic approximation. The vector potential for the field of a single loop of radius Rc in cylindrical coordinates has the form 12⁄ µI R c 2 2 2 Aθ =------[]()1Ð 0.5k Kk()ÐEk(),(1) πkr Rc R0 Rw R with B = ∇ × A and the magnetic flux surfaces are given by rAθ. K and E are complete elliptic integrals of the Fig. 2. Midplane pressure profile shown schematically. first and second kind. The ring current is I and 4Rr k2= ------c- . Outside of the peak pressure region the pressure will ()R+r2+z2 decay in a region of “bad” curvature and MHD stability c requires a relatively gentle pressure gradient. This For r ӷ Rc the field has a dipole dependence results in a large volume of low field and of low density µM 12⁄ plasma. Thus a dipole confinement device would look Br(),θ =------0()13+ sin2 θ . (2) like a small ring surrounded by a hot plasma that is cen- 4πr3 tered within a relatively large spherical vacuum chamber. Beta, the ratio of plasma to magnetic field pressure An additional vertical field term, Aθ⊥ = 0.5B⊥r term is a measure of the efficiency of the utilization of the must be added to the vector potential (Eq. 1) to produce magnetic field and since the field and pressure fall off a diverted geometry. together, β only decays slowly with radius. When a small uniform field is added in the direction 3. SCRAPE-OFF LAYER CONSTRAINT of the dipole axis a separatrix will form which leads to two possible geometries: (1) When the uniform field Since the pressure gradient must not exceed a criti- adds to the outer dipole field the separatrix exhibits two cal value the peak pressure, p0, is determined by the symmetric nulls on the dipole axis as exhibited in scrape-off layer pressure: Fig. 3. (2) When the uniform field subtracts from the ⁄ ⁄ ()⁄ 20 3 ∼ 5 7 outer dipole field the separatrix exhibits a field null on p0 psol = rsol r0 10 Ð10 . the outer midplane, as seen in Fig. 4. Within the last Assuming that a fraction f of power leaving the closed flux surface we expect to satisfy the MHD R δ γ plasma is radiated we can balance the power loss from requirement (pV ) > 1. Outside the last closed flux sur- the plasma with the flow in the scrape-off layer: face the plasma will flow along field lines into the end () wall. 1 Ð f R p0V0≈ ------τ- 2 p solAsolcs, Beyond the separatrix heat will be conducted along E the field lines and flow toward the end plates or limiter. A strong fanning of the diverted field lines is expected with cs the scrape-off layer sound speed, V0 the effec- to greatly reduce the end wall power density. The tive volume of hot plasma defined by p V = pdV, A plasma at the divertor plates is expected to be 1 to 0 0 ∫ sol 10 eV and at the symmetry point (opposite the end the scrape-off layer cross-section area and p0 (psol) the plate) 20 to 100 eV. As with a tokamak divertor the respective core (scrape-off layer) pressures. The zero scrape-off layer plasma can be lead into a baffled cham- subscript denotes values at the location of the pressure ber to permit pumping at high neutral pressure with a peak. Assuming that the scrape-off layer width is a ≈ πγ ρ limited back streaming of the neutralized outflow. finite number of ion gyro radii, we take Asol Rsol sol ρ γ One straightforward way to heat a laboratory dipole with sol the scrape-off layer ion gyro radius and the experiment would be to utilize electron cyclotron reso- ratio of the scrape-off layer width to the ion gyro radius nance heating (ECRH). For an ECRH heated experi- γ ≥ ≈ π 3 ( 1). Taking V0 R0 and imposing the critical tem- ment the heating will take place at both the fundamen- ∝ –8/3 γ tal and the second harmonic resonance field locations perature profile T R and = 1 we obtain and the hot electrons will be localized between the flux 2 16⁄ 3 crit BRR surfaces that are tangent to the fundamental and the τ ≈ 0.5() 1 Ð f ------00------sol- (3) E R R second harmonic mod-B surfaces. A typical laboratory Ti0Te00 embodiment of a levitated dipole could utilize 2.45 to 28 GC ECR heating. The coil design used in Fig. [1] is in MKS units with temperatures in eV. Consider two constrained by the current density limit for a supercon- examples: 8 2 ductor, taken here to be 1 × 10 A/m , and by the 1. A dipole with R0 = 0.4 m, Rsol = 3 m, B0 = 0.4 T, requirement that the resonant flux tube clears the coil. fR = 0.5. For a hot electron plasma with Te0 = 300 KeV,

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 PLASMA CONFINEMENT IN A LEVITATED MAGNETIC DIPOLE 745

Z, cm Z, cm –300 –300 5 × 10 –3

–3 × 10 5 –200 –200 1 × 10

–2

1 1

10 × –2

–100 –100

2 2

× ×

0 10 0 10

–1 –1

5 5 5

10 10 × ×

–1 –1

1 1 1

× × 10 10 –1 100 –1 100

200 200

300 300 0 100 200 300 400 0 100 200 300 400 R, cm R, cm

Fig. 3. Conﬁguration with separatrix null points in dipole Fig. 4. Conﬁguration with separatrix null on outer mid- axis. plane.

τ and Ti0 = 200 eV, we obtain E ~ 0.1 sec. For a density ior results from the constraint imposed by the adiabatic × 1 –3 β of ne = 5 10 m this corresponds to e = 0.38 and the profiles of pressure and temperature. power requirement of 75 KW. This estimate of power 8/3 ignores the stabilization of the hot electron interchange We have assumed that Tsol/T0 = (R0/Rsol) . More mode by the thermal electron component [12] which generally the temperature profile is determined by the would permit a higher ratio of peak to scrape-off layer heating and thermal transport profiles. The relationship pressure and thereby reduce the power requirements. of the temperature to the density profile is also con- Therefore this heating power estimate may be consid- strained by stability considerations as discussed below. ered to be an upper bound. Furthermore the scrape-off layer temperature is determined by the cooling processes that take place within 2. For R0 = 0.4 m, Rsol = 3 m, Ti0 = Te0 = 5 KeV, B0 = the scrape-off layer including radiation and secondary τ 0.4 T, f = 0.5 we obtain E = 0.15 sec. For a density of electron emission. As the plasma flows along the × 19 β ne = 2 10 this corresponds to = 0.5 and the power scrape-off layer field lines toward the end plates it will required to sustain this plasma is 65 KW. cool and it can become collisional. The scaling changes when the mean free path within the scrape-off layer τ τ crit For E < E from (3) the scrape-off layer can becomes less than the connection length and the heat expand to accommodate a higher heat flow, i.e. γ > 1. flow along the field lines becomes predominantly con- ductive. For a collisional plasma the temperature will τ τcrit For E > E the heating power must be decreased, i.e. vary along the field line in the scrape-off layer (at con- the temperature T0 must decrease for the heating power stant pressure) due to parallel thermal conductivity. to balance the scrape-off layer power outflow with the Additionally if the neutral pressure becomes suffi- divertor width fixed at one ion gyro radius. This behav- ciently large (in the vicinity of the end walls) the pres-

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 746 KESNER, MAUEL sure can fall due to momentum exchange with neutral Interchange Modes from Kinetic Theory particles. ψ When F0 = F0(⑀, ) the kinetic analysis of electro- For a collisional scrape-off layer the power balance static drift modes proceeds from a traditional approach becomes [16] beginning with the drift kinetic equation. A kinetic stability analysis of interchange modes produces simi- ()1 Ð f pVp χ||V lar results to the MHD interchange criterion. In [16] it ------R 0-0≈ ------sol sol, τ 2 is shown that, interchange modes become stable when E Lsol the following criterion is satisﬁed:

7/2 3 2 and we note that for classical conductivity χ|| ∝ T . v ()ω ω ω > ∫d F0 d Ð d * 0, (4) Since collisionality retards the outflow of heat along ω × ∇ Ω open field lines the scrape-off layer temperature can be where ∗ = b k⊥ á 'F0/m cF0⑀ is the diamagnetic higher without degrading core confinement as com- ω × 2 ∇ µ∇ Ω pared with a collisional scrape-off layer. drift frequency, d = k⊥ á b (mv || b á b + B)/m c is the processional drift frequency, and b = B/|B|. In ω ω ω this analysis we have assumed Ӷ b with b the ω 4. MHD AND ELECTROSTATIC MODE bounce frequency defined by b = v||d/ds and we have STABILITY bounce averaged the drift kinetic equation. The bounce averaged quantities appear with an overline, a = The dipole reactor concept is based on the idea of generating pressure profiles near marginal stability for ∫a dl/v||. This yields the approximate stability criterion low-frequency MHD and electrostatic fluctuations. ω > (1 + η)ω∗ which indicates that (at low β) stability From ideal MHD, marginal stability results when the d pressure profile, p(ψ) satisfies the adiabaticity condi- requires that the local pressure scale length exceeds the tion, δ(pVγ) = 0, where V is the specific flux tube vol- radius of curvature. ume and γ = 5/3 [13]. Hasegawa pointed out that when the invariants µ and J are conserved, marginal stability Trapped Particle Modes of both MHD and drift modes results when ∂F(µ, J, ψ ∂ψ Trapped particle modes are low frequency electro- )/ = 0, where F is the particle distribution function ω ω Ω and ψ is the third adiabatic invariant. Both of these con- static modes ( Ӷ b Ӷ c) that can localize in the ω ω ≈ ditions lead to dipole pressure profiles that scale with outer part of the torus. For a tokamak d/ ∗ 1/A Ӷ 1 radius as p ∝ r–20/3. and as a result we can ignore resonances with the pre- ω cessional drift, d, and obtain drift frequency fluctua- At high beta MHD ballooning is expected to provide tions of order ω ~ ω∗. With a dipole we have the oppo- a stability limit. The stability of ballooning modes has ω ω been studied for both magnetospheric and fusion dipole site ordering, i.e. d/ ∗ > 1 and stability of interchange ω η ω applications [14Ð15] and these studies indicate a beta modes require d > (1 + ) ∗. The dipole ordering limit in excess of β > 1. For application to planetary leads to the result that instability may be driven by a systems or for supported dipole experiments one must resonance of the wave with the processional drift ω ω take account of anisotropic pressure and also the motion, i.e. ~ d. boundary conditions where the field lines enter the poles. For fusion applications one is interested in a lev- An analysis of the dissipative trapped ion mode that assumes collisional electrons and collisionless ions itated dipole which has closed field lines and therefore ω ηω we expect the pressure to be isotropic. For an isotropic indicates a stability criterion d > ∗/2 [16]. Like- pressure we can apply the standard MHD formalism. wise an analysis of a collisionless trapped particle ω ηω mode indicates a stability criterion d > ∗ [16]. On the other hand, since confinement in a levitated Therefore we can conclude that, in a levitated dipole, a dipole results from cross field transport, (and not pitch- pressure profile that is stable to interchange modes may angle scatter) it may be expected that, on a transport also be stable to trapped particle modes. time scale, the distribution function will become isotropic. To lowest order the distribution function would be 2 2 η ψ i Modes approximated by F0 = F0(⑀, ) with ⑀ = (v ⊥ + v || )/2, η the particle energy. It still remains true that low fre- In a tokamak the so-called i or “mixing” modes are quency unstable modes (ω Ӷ ω , Ω ) would lead non- electrostatic modes that couple ion acoustic and drift b c ω Ω linearly to the adiabatic distribution function, ∂F(µ, J, fluctuations. They are low frequency modes Ӷ ci ψ)/∂ψ = 0. The subsequent collisional relaxation of the but include the dynamics of the bounce frequency time distribution function will preserve n, T, and η (η ≡ scale. A dipole does not have magnetic shear and so we η dlnT/dlnn) and this has important consequences for will refer to the early derivations of the i mode which micro stability. do not include magnetic shear. Antonsen et al. [17Ð18]

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 PLASMA CONFINEMENT IN A LEVITATED MAGNETIC DIPOLE 747 derives stability criteria from both a fluid and a colli- means for the adiabatic convection of the fusion prod- sionless kinetic theory. The simplest fluid theory indi- ucts. One might then consider the possibility that, in a η cates that instability occurs when i > 2/3. The adia- reactor, a fraction of the synchrotron radiation that is η ∇ ∇ batic distribution is characterized i = lnTi/ lnn = emitted from the plasma can be reflected back so as to 2/3. The more detailed fluid derivation and the colli- be non-uniformly reabsorbed and therefore drive con- sionless kinetic approach indicate that instability vective cells that fuel and cleanse the plasma. η occurs when i > 1. Thus adiabatic profiles which are η characterized by i = 2/3 are expected to be stable to these modes. Conclusions on Stability In general the radial profiles of heating and fueling Drift Cyclotron Modes determine the respective temperature and density profiles. We can imagine heating and fueling profiles that Drift cyclotron modes are high frequency unstable generate marginally stable pressure profiles (p ∝ V–γ) modes (ω ~ Ω ) modes that are driven by temperature η ci while maintaining i < 2/3. If we exceed the critical and density gradients. Pastukhov and Sokolov have pressure gradient we would expect the plasma to evaluated the transport from these modes in the good expand unstably so as to broaden the pressure profile. If curvature region near the surface of the levitated dipole η we exceed a critical value of i we would expect an [19, 20]. They show that the resulting transport would onset of micro-turbulence drive transport. This is in be severely limited by particle recycling at the surface direct analogy to the operation of a tokamak. With pres- of the internal coil. Because the surface of dipole is η sure gradients and i appropriately bounded a levitated completely surrounded by a dense plasma, the net par- dipole may exhibit classical transport. ticle flux to the ring must vanish. A cool, high-density sheath forms at the dipole surface which transforms the thermal flux into bremsstrahlung radiation. 5. ECR HEATING AND PELLET INJECTION Electron cyclotron resonance heating provides a Stability of Convective Cells straight forward method to build up a substantial stored Experiments in multipoles have indicated that con- energy in a hot electron plasma. This capability results vective cells [21] can provide the dominant source of from the predicted stability at high beta and the closed cross-field transport in shear-free systems. It is under- field line geometry of a levitated dipole. The energy stood theoretically [22] that zero frequency convective stored in the hot electron plasma can be transferred into cells are closely related to interchange modes and they a high density, thermal hydrogenic plasma by injecting will grow in regions of bad magnetic well, i.e. where pellets into the ECR heated hot electron plasma. The δ(pVγ) < 0. In addition it has been shown that convec- energy transfer into the hydrogenic and the thermal tive cells will exist in regions of good curvature when electron species depends on a competition between re- the heating is non-uniform [23]. In the Wisconsin octo- thermalization and radial thermal transport. If transport pole experiments convective cells were observed in is classical or near classical essentially all of the stored regions of both good and bad curvature [22] regions energy can be transferred into the hydrogenic and the and the convective cells were observed to decay. In thermal electron species. In this optimistic scenario these experiments the initial plasma was non-uniformly D3He ignition could be obtained in an inexpensive distributed within the chamber. These experiments also manner in a relatively small reactor. indicated that a small amount of shear caused a rapid decay of the convective cells. Additionally it was If we consider, for example, a dipole plasma heated observed that small field errors can cause convective by a 28 GC gyrotron at the 1 T resonance surface (about flow patterns in a shear-free configuration [24]. 100Ð200 KW of power) we may expect to produce a × 18 –3 hot electron plasma with ne ~ 2–5 10 m at 200Ð These results lead one to suspect that uniform heat- 500 KeV (0.2 < β < 1). The ability to transfer the energy ing will be required in a levitated dipole in order to depends on the competition of cross field transport and avoid the excitation of convective cells. Alternatively thermalization of the hot electron energy. For classical one can consider the addition of a small current on the confinement, a large fraction of the energy could be dipole axis (or appropriate current drive) to create some × 18 –3 shear and thereby eliminate these modes. transferred and a ne ~ 2 10 m , 200 KeV electron plasma would yield a 10 KeV hydrogenic plasma at On the other hand, convective cells can be usefill in n ~ 2 × 1019 m–3. a reactor to purge the ash. At marginal stability, pVγ = e const, the exchange of flux tubes will transport particles To study this process we utilize a zero dimensional ∝ η but not energy. Additionally when ne 1/V ( i = 2/3) model assuming classical re-thermalization between the exchange of flux tubes will not change the density the suprathermal and the thermal species. Thermal profiles. Therefore for an adiabatic profile the con- plasma losses due to cross field transport characterized τ τ trolled application of asymmetric heating can provide a by E and P. We solve the following simultaneous rate

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 748 KESNER, MAUEL

T, eV T T, eV e 3500 12 000 Te 10 000 3000 8000 2500 2000 6000 T i 1500 4000 1000 Ti 2000 500 0.2 0.4 0.6 0.8 1.0 t, c 0.1 0.2 0.3 0.4 0.5 0.6 t, c Fig. 5. Solution to rate equations for electron and ion tem- 2 peratures with τ = a /χ⊥ for a = 0.1 m, and initial values of E Fig. 6. Solution to rate equations for electron and ion tem- n = 5 × 1018 m–3, n = 5 × 1019 m–3. τ τ eh i0 peratures with E = 20 ms, p = 200 ms and a = 0.1 m.

equations for hot electrons, Teh, thermal electrons, Te, 6. THERMAL EQUILIBRIA OF AN IGNITED 3 thermal ions, Ti, and density, ne (assuming ne = ni): D HE PLASMA An estimate of the equilibrium profiles in an ignited dTeh T e Ð T eh T i Ð T eh ------= ------+ ------, D3He plasma requires a solution of a heat transport dt τ τ ei ih equation. This is a two-dimensional problem since the magnetic field strength varies along a field line (the dTe T eh Ð Te T i Ð T e T e density and temperature are flux functions) In a levi------= ------τ - + ------τ - Ð τ------, dt ei ei Ee tated dipole geometry a major portion of the volume of confined plasma is located near the outer midplane. We dT T Ð T T Ð T T can therefore estimate confinement using a one dimen------i = ------eh i Ð ------i -e Ð ------i , dt τ τ τ sional model that uses fields evaluated at the outer mid- ih ei Ei plane of the dipole. We assume D3He power production with a 20% D, 80% 3He mixture and furthermore dn n i i assume that the dominant loss channels are ------= Ð τ----- , dt p Bremsstrahlung radiation and classical transport. The magnetic field also enters the Jacobian to describe the τ τ with ei the classical rethermalization time, ih the hot flux tube expansion and for classical transport κ⊥ ∝ τ τ electron-ion rethermalization time, Ei( Ee) the ion 1/B2. We can write the thermal transport equation in τ (electron) energy confinement and p the particle con- flux coordinates using the magnetic scalar potential, χ, finement time. The energy confinement time is related as the angular coordinate, i.e. B = ∇χ. The flux tube τ ≈ δ2 χ to a mean transport coefficient by E /4 . average of the transport equation then gives: For classical confinement essentially all of the ∂ ∂ ()ψ [] ∂χ2κ T energy can be transferred into the thermal plasma. One ∂ψn ∫R ⊥∂ψ------example would be to consider classical confinement (5) τ 2 χ dχ with E = a / ⊥ for a = 0.1 m and initial values of neh = []〈〉σ () () = nDnHe v 3 E fus T ÐPbrem T ∫ ------. × 18 –3 × 19 –3 DHe 2 5 10 m , ni0 = 5 10 m . As shown in Fig. 5 Te B rises up to 12 KeV and Ti reaches 8 KeV. A second 〈σ 〉 3 example, shown in Fig. 6 shows the solution to rate On the right hand side v 3 is the D He rate coef- τ τ D He equations for with E = 20 ms p = 200 ms and a = ficient, Efus = 18.3 MeV and Pbrem is the Bremsstrahlung 0.1 m. Even for substantially reduced ion confinement radiation power given in [25]: significant plasma parameters would be attained due to energy transfer from the relativistic electrons. Thus, by × Ð37 2[ ( × Ð3 combining ECRH and pellet injection a modest dipole P brem = 510 Tne Zeff 1+ 1.55 10 T e ⁄ experiment might be capable of investigating the prop- + 7.15×) 10Ð6T 2 +0.071()ΣZ 3n ⁄ n TÐ12 erties of high temperature, high density, and high beta e i i e e ] plasmas. + 0.00414T e

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 PLASMA CONFINEMENT IN A LEVITATED MAGNETIC DIPOLE 749

3 –3 3 in W/m with T in KeV and ne in m . On the dipole shield and structure exceeds 5000 m ). The dipole reac- midplane R = R0 and (4) becomes: tor concept also differs from the spherator [26] since the plasma profiles of the spherator are steep (i.e. they 1 ∂ B 0 ∂ l 2 ∂ T cannot be made stationary) and low-frequency fluctua------n κ ⊥ ------R ------R B ∂ R e 0 R ∫ B ∂ R tions or convection cells may significantly degrade con- 0 0 0 0 0 (6) finement. []〈〉σ () () dl = nDnHe v 3 E fus T ÐPbrem T ∫ ----- . Conceptual dipole reactor designs have been DHe B reported [3Ð4], and the use a dipole fusion reactor for 2 space propulsion has been proposed [5]. In each of We have used κ⊥ ∝ 1/B i.e. we have assumed classical −3 transport. these designs, D He fuel was used instead of the more highly reactive DÐT fuel in order to reduce the neutron The flux average integrals, can be approximated by flux to the levitated coil. Also reflectors were used to ()dl⁄ B R2 ∝ R6 and ()dlB⁄ ∝ R4. The scaling of reduce synchrotron losses from the high-pressure and ∫ 0 ∫ 0 β ∆ lower plasma on the inside of the levitated dipole. (5) indicates that the temperature half-width, T scales The high β capability of the dipole reactor makes pos- ∆ ∝ κ⁄ () sible the use of advanced and possibly aneutronic fuels, as T ⊥HT with H(T) the sum of the heating terms from the right hand side of (5) Therefore a larger but the high temperatures required to burn these fuels thermal transport leads to a broader temperature profile necessitate steps to reduce synchrotron emission losses. If we compare the total heat flux toward the ring vs The designs reported in [3] and [4] described compact toward the scrape-off-layer we can show that when and relatively low-power dipole reactors with large ∆ Ӷ R with R the location of the temperature peak. plasma expansion regions. A 20 MA dipole coil of T 0T 0T radius of 1.8 m confined a plasma with peak β 3 and The ratio of heat to the ring vs heat to the scrape-off- ~ generated 100 MW of fusion power A higher field, layer is H H ≈ ∆ R . R/ sol 1 + 6 T/ 0T 40 MA dipole with a denser plasma at the same β could To solve Eq. (6) a density dependence must be generate 1000 MW. The plasma is heated to ignition assigned but since all terms go as n2 this dependence with direct heating of the plasma core (using, for exam- only has a weak effect on the solution. (The pressure ple, neutral beam injection). In [5], a much larger gradient, however, enters into the stability criteria and dipole reactor containing a 54 MA dipole having a the pressure gradient broadens when the density profile radius of 6 m and producing 2000 MW of fusion power broadens). The solution of (5) is a strong function the was considered for rocket propulsion The plasma amplitude and radial dependence of κ⊥ which we have expansion region was not as large and a relatively hot taken to be classical. Thus at a fixed beta, as the mag- plasma was diverted to an annular gas-neutralizer to netic field increases the peak fusion power density rises generate thrust In both [4] and [5], thermoelectric con- as B2 but additionally the hot plasma region narrows verters were located within the levitated dipole, and due to reduced transport. they provided the power to drive refrigerators for the superconducting magnets. Designs of the superconducting magnets and shields in [3] and [5] illustrate the 7. DISCUSSION feasibility of reactor-sized dipole magnets using present-day multi-filamentary Nb Sn conductors. A dipole fusion reactor would consist of a single 3 levitated circular magnet within a large vacuum cham- Closely related studies of plasma confinement in ber. The hot plasma core would encircle the levitated multipole fields have been a subject of significant dipole coil forming a toroidal annulus. A large expan- research efforts [27, 28]. Recent theoretical work sion region of cooler plasma extends outward from the includes the study of the equilibrium of two dimen- dipole where the plasma pressure decreases with tional internal coil configurations [27–30]. 20/3 radius, R, approximately as R characteristic of the Finally, a supported dipole laboratory experiment at marginally stable profiles found in magnetospheres. Columbia University has studied the detailed phase- Although the overall dimensions of the dipole fusion space evolution of dipole-trapped energetic plasma in reactor may be large, the size of superconducting the presence of intense drift-resonant fluctuations [9– dipole magnet is small. Indeed, in the dipole reactor 10]. In this experiment, an “artificial radiation belt” (a conceptual designs [3Ð5] the volume of the hot plasma population of 5Ð50 KeV energetic electrons) is pro- core (40 m3) exceeds the volume of the levitated ring duced with microwave heating which excites hot-elec- (20 m3). This feature of the dipole reactor (i.e. a larger tron interchange instabilities when the pressure gradi- plasma volume than the volume of the high-technology ent sufficiently exceeds the marginally stable profiles superconducting magnets, shield and structure) is in envisioned for the dipole reactor. Fluctuations leading sharp contrast to the tokamak where the volume of the to global chaotic transport as well as thin, localized plasma is usually less than the volume of the surround- regions of stochastic drifts are observed, and these ing fusion island. (For example, in ITER, the plasma observations have been used to verify models of colli- volume is 2500 m3 and the volume of the magnets, sionless radial transport in dipole magnetic fields.

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997 750 KESNER, MAUEL

REFERENCES 16. Kesner, J., Phys. Plasmas, 1997, vol. 4, p. 419. 1. Hasegawa, A., Comments on Plasma Physics and Con- 17. Coppi, B., Rosenbluth, M., Sagdeev, R., Phys. Fluids, trolled Fusion, 1987, vol. 1, p. 147. 1967, vol. 10, p. 582. 2. Vampola, A.L., Symp. on Natural and Manmade Rad. in 18. Antonsen, T., Coppi, B., Englade, R., Nuc. Fus., 1979, Space, (Wash., D.C.), 1972, p. 538. vol. 19, p. 641. 3. Hasegawa, A., Chen, L., and Mauel, M., Nuclear Fus., 19. Pastukhov, V. and Sokolov, A.Yu., Nuc. Fusion, 1992, 1990, vol. 30, p. 2405. vol. 32, p. 1725. 4. Hasegawa, A., Chen, L., Mauel, M., Warren, H., 20. Sokolov, A.Yu., Plasma Phys. Rep., 1993, vol. 19, Murakami, S., Fus Tech., 1992, vol. 22, p. 27. p. 1454. 5. Teller, E., Glass, A., Fowler T.K., et al., Fusion Technol- 21. Dawson, J.M., Okuda, H., and Carlile, R., Phys. Rev. ogy, 1992, vol. 22, p. 82. Lett., 1971, vol. 27, p. 491. 6. Dawson, J.M., Private Communication. See [5]. 22. Hassam, A. and Kulsrud, R., Phys Plasmas, 1979, vol. 22, p. 2097. 7. Mauel, M., Warren, H., Hasegawa, A., IEEE Transac- 23. Hassam, A., private communication, 1997. tions on Plasma Science, 1992, vol. 20, p. 626. 24. Jernigan, T., Rudmin, J., and Meade, D., PRL, 1971, 8. Mauel, M.E., Hasegawa, A., in Physics of Space Plas- vol. 26, p. 1298. mas, Cambridge Scientiﬁc Publs., 1990, p. 461. 25. McNally, J.R., Nuclear Technology/Fusion, 1982, vol. 2, 9. Warren, H.P. and Mauel, M.E., Phys. Rev. Lett., 1995, p. 9. vol. 74, p. 1351. 26. Freeman, et al., “Conﬁnement of plasmas in the spher- 10. Warren, H.P. and Mauel, M.E., Phys. Plasmas, 1995, ator”, in Proc. 4th Conf. Plasma Physics and Controlled vol. 2, p. 4185. Nuclear Fusion, Madison, Wisconsin, June 17Ð23, 1971, 11. Sackett, S., “EFFI Electromagnetic Field Code”, LLNL IAEA-CN-28/A-3, vol. 1, p. 27. Report UCRL-52402, 1978. 27. Ohkawa, T. and Kerst, D.W., Nuovo Cimento, 1961, 12. Berk, H.L., Phys. Fluids, 1966, vol. 19, p. 1275. vol. 22, p. 784. 13. Rosenbluth, M.N. and Longmire, Ann. Phys., 1957, 28. Navratil, G., Post, R.S., and Butcher Ehrhardt, A., Phys vol. 1, p. 120. Fluids, 1977, vol. 20, p. 157. 14. Hameiri, E., et al., J. Geophys. Res., 1991, vol. 96, 29. Maikov, A.R., Morozov, A.I., and Sveshnikov, A.G., p. 1513. Plasma Physics Reports, 1996, vol. 22, p. 87. 15. Chan, A., Xia, M., and Chell, L., J. Geophys. Res., 1994, 30. Morozov, A.I. and Murzina, M.B., Plasma Physics vol. 99, p. 17351. Reports, 1996, vol. 22, p. 500.

bremsstrahlung useﬁll

PLASMA PHYSICS REPORTS Vol. 23 No. 9 1997