1. GENERAL FRAMEWORK
The purpose of this Section is to provide the general framework of the research on compact tori. As compact tori are here designated all the magnetic confinement configurations in which a closed magnetic surface takes the shape of a torus with the minor radius a approaching the major radius R: i.e. whose aspect ratio tends to unity, A=R/a®1.
The main configurations included within this framework are spherical tori (ST), spheromaks and field reversed configurations (FRC). The spherical torus is at the moment the one explored with most success, due to its similarity to the much more investigated tokamak confinement scheme. As a matter of fact one could argue that the spherical torus is the attempt of solving many of the tokamak problems (turbulence, disruptions, beta limit, etc ) by pushing upon the configuration leverage. On the other hand, the latter two confinement schemes (spheromaks and field-reversed configurations) have been much less studied in the laboratory, although they possess in principle many attractive features.
The reason why they have, up to now, been less successful is mainly connected with the fact that they rely more heavily upon plasma self-organization, both for their formation as well as for their sustainment. Although many formation schemes have produced in the last twenty years interesting spheromaks and field reversed configurations, at the present moment (July 2001) no sustainment scheme has been soundly and fully demonstrated.
The PROTO-SPHERA experiment aims at sustaining a flux-core-spheromak, while exploring the configuration space that connects spherical tori and spheromaks. The compression of the central pinch, while decreasing the total longitudinal pinch current, would lead, if successful, to the formation of a field reversed configuration. So PROTO-SPHERA could also explore a new technique for setting up an FRC. Finally PROTO-SPHERA could also aim at exploring the novel Chandrasekhar-Kendall-Furth configuration consisting of a spherical torus enclosed within a spheromak, which will be introduced in Section 3.
1.1. Spherical Tori, Spheromaks and FRCs
After more than thirty years of development, the tokamak concept has come very close to achieving controlled thermonuclear fusion break-even conditions and a number of proposed next generation experimental devices could provide a burning plasma. However, the tokamak is very large, complex, and expensive. Even improved tokamaks may not overcome the shortcomings of low power density, high complexity, large unit size, and high development cost. It is therefore important to develop alternatives to conventional tokamaks with designs optimized for simplicity, small size and low cost. Several alternatives have been proposed with various tradeoffs between ultimate attractiveness and present feasibility.
A number of research groups world-wide have been working on three related alternate concepts: the spherical torus (ST), the spheromak, and the field-reversed configuration (FRC). These concepts are at very different stages of development.
Fig. 1.1 compares the magnetic topologies of ST, spheromak and FRC. The ST is a modification of the conventional tokamak and differs by having a much smaller aspect ratio. Spheromaks are low b toroidal confinement configurations where currents flowing in the plasma produce the magnetic field almost entirely; they have a finite internal toroidal magnetic field, which vanishes at the plasma surface; hence no external field coils link the plasma. FRCs are high b toroidal confinement configurations with zero toroidal magnetic field everywhere and so, like spheromaks, do not have coils linking the plasma. Thus, spheromaks manage to have a toroidal field without having toroidal field coils; FRCs do not have toroidal field coils, but also do not have a toroidal field.
Fig. 1.1. Comparison between ST, spheromak and FRC.
The spheromak [1, 2] is a compact magnetofluid configuration of simple geometry with attractive reactor attributes, including no material centerpost, high engineering beta, and sustained steady-state operation through helicity injection. It is a candidate for liquid metal walls in a high-power-density reactor and has a simple geometry for incorporating a divertor, as shown schematically in Fig. 1.2. It has toroidal and poloidal fields of comparable strengths.
Fig. 1.2. Schematic of a self-ordered spheromak configuration, illustrating near
spherical reactor geometry using liquid metal blanket and shield.
Spheromaks do not use a transformer (as in tokamaks) to produce the nested poloidal flux surfaces required for confinement. Instead spheromaks are formed by the self-organization of naturally occurring MHD instabilities. The self-organization means that there is not a unique way to make spheromaks, and indeed, several different methods have been successfully demonstrated [3, 4, 5 and 6]. Magnetic helicity (linked magnetic fluxes) plays an important role in forming and sustaining spheromaks. An initial configuration with sufficient helicity and energy will spontaneously relax to a spheromak, given appropriate boundary conditions. Figure 1.3 shows how a magnetized coaxial plasma gun creates a spheromak. A puff of gas is introduced into the annular gap between the inner and outer coaxial cylindrical electrode  (Fig. 1.3.a). High voltage capacitors charged to 5÷10 kV are connected to the electrodes and cause the gas to ionize and become a toroid of plasma. The current flowing in the gun and through the plasma interacts with its own magnetic field to produce a force, which accelerates the plasma towards the open end of the gun (Fig. 1.3.b). A strong magnetic field, called the "stuffing field", is produced by an external magnetic coil and is concentrated in the center electrode with a slug of high permittivity metal. The plasma encounters this magnetic field at the opening of the gun and resists the change in field, according to Faraday's law. Because the plasma is an excellent conductor, currents flow in the toroid of plasma as it distends the stuffing field (Fig. 1.3.c). If the magnetic pressure from the gun exceeds the magnetic tension of the stuffing field, the toroid breaks away to form a spheromak. The field lines distend and then reconnect in back as the spheromak forms. The spheromak inherits toroidal field from the gun field and poloidal field from the stuffing field (Fig. 1.3.d). The amount of gun current required to overcome the stuffing field is called the formation threshold.
Fig. 1.3. Spheromak formation: injection by a plasma gun inside a flux conserver .
In the exploratory scale device CTX (Los Alamos National Laboratory) central electron temperatures Te(0)=400 eV, average b~5% and central b(0)~20% were obtained with a 2-T magnetic field . Analysis of CTX data found the energy confinement in the plasma core to be consistent with Rechester-Rosenbluth transport in a fluctuating magnetic field, potentially scaling to good confinement at higher electron temperatures.
The MHD stability against the tilt mode is an issue as well as the efficient sustainment of the plasma current. Electrostatic helicity injection has been demonstrated to sustain the spheromak current via a magnetic dynamo involving flux conversion and has been implemented, for limited duration, in several experiments. Experiments have shown that the spheromak is subject to continuous resistive MHD modes, similar to those in the reversed field pinches (RFP), which tear the magnetic fields but reduce the plasma confinement. The SPHEX group (Manchester, England) studied the dynamo in sustained spheromaks in a cold plasma . A new concept exploration experiment , SSPX (see Fig. 1.4), has recently begun operation at Lawrence Livermore National Laboratory and is addressing the physics of a mid-sized sustained spheromak with tokamak-quality vacuum conditions and no diagnostics internal to the plasma.
Fig. 1.4. Schematic of the new SSPX spheromak experiment at Livermore.
Operating spheromak experiments are the Swarthmore Spheromak Experiment (SSX) (reconnection) , the Caltech Helicity Experiment (spheromak formation issues) , the FACT/HIST experiment at the Himeji Institute of Technology  and BCTX at the University of California, Berkeley. The physics of reconnection is being studied in MRX at Princeton Plasma Physics Laboratory (PPPL)  and on TS-3 and TS-4 at the University of Tokyo .
A field-reversed configuration (FRC) is a compact torus plasma with negligible toroidal magnetic field. It is usually fairly elongated, contained in a magnetic field produced by a cylindrical solenoid, and possesses a simple, unobstructed divertor (Fig. 1.5). The plasma beta is close to unity, and an FRC is thus both extremely compact and geometrically simple [15, 16]. The coils and divertor geometry are the simplest of any configuration.
Fig. 1.5. FRC geometry.
FRCs have been formed with high plasma pressures in theta-pinch devices , like BN (Triniti), FIX (Osaka University), NUCTE-3 (Nihon University), LSX, STX and TCS (University of Washington).
Fig. 1.6. Schematic drawing of the FIX machine, with axial B0 profile in vacuum .
Outside a quartz vessel the theta-pinch coils (see Fig. 1.6) are typically pulsed to 1 T field level in a few ms. Without an external current drive, these current rings decay on sub-millisecond L/R times. Typical electron densities of ne~51021 m-3, ntE~1018 m3·s and the highest average b of 50% to 80% have been achieved in FRCs with major radii of 15 cm, at several 100-eV temperatures. Lifetime has been observed to increase with density: shorter-lived FRCs are easily produced at ne~1021 m-3, with keV temperatures.
After the impulsive formation inside a theta-pinch coil the FRC is translated along a guide field, expanded (lowering ne by factors up to 100) inside a metal vessel with quasi-steady magnetic field and then stopped by a mirror field (see Fig. 1.7).
Fig. 1.7. Time evolution of separatrix radius profile rs and of flux function in FRX .
The theta-pinch formation technique is limited to the tens of mWb level. Several Weber are required for a reactor, so other methods of formation are being studied. The slow formation of an FRC, using two merging spheromaks with opposite helicity  (Fig. 1.8) has been demonstrated  on spheromak merging facilities, such as TS-3 (Tokyo), MRX (Princeton) and SSX (Swarthmore), see Fig. 1.9.
Fig. 1.8. Calculated evolution of counter-helicity spheromaks merging into an FRC .
Fig. 1.9. TS-3 spheromak merging facility and experimental results from magnetics .
A promising approach for FRC sustainement is the application of a rotating magnetic field (RMF), using large antennas (see Fig. 1.10).
Fig. 1.10. Schematic of FRC sustainement by RMF .
A partially ionized spherical FRC discharge has been produced and sustained for 40 ms by (RMF) in the Rotamak device  (Flinders University, Australia).
An interesting observation is that the FRC plasmas produced in all the experiments are more globally stable than ideal MHD theory would predict. The observed stability in present experiments is thought to be due to kinetic effects, which have been characterized by a parameter s, equal to the number of ion gyro-radii between the field null R and the separatrix rs (see Fig. 1.5). The utility of the concept depends upon demonstrating stability as s is increased from present values of about 4 to the level of 20÷30 thought necessary to provide reactor level confinement. The enhancement of kinetic stabilization, either through addition of energetic particles (e.g., ion ring merging or neutral beam current drive) or naturally occurring fusion reaction particles, may be an essential component of the concept. Furthermore there is some theoretical and experimental evidence that FRCs may be naturally occurring minimum energy states stabilized by sheared rotation , akin to spheromaks and reversed-field-pinches (RFP), when the total helicity (which includes angular momentum) is conserved.
Among alternative concepts based on low-density plasma magnetic confinement, the FRC offers arguably the best reactor potential because of high power density, simple structural and magnetic topology, simple heat exhaust handling, and potential for advanced fuels. Particle distributions driven, for example, by beams and including the effects of nuclear polarization can provide certain benefits in magnetic fusion devices. Beams of ions, colliding at energies near the peak in their fusion cross-section, lead to a higher Q than a thermal distribution of the same mean energy. This increase may be in the form of a nuclear resonance; hence, such distributions are far better than simply hotter plasmas. The benefits and needs are greatest for high-beta magnetic fusion energy devices, for example, the field-reversed configuration (FRC), spheromak, and spherical tokamak (ST). Because of its demonstrated very high beta and potential for direct electrical conversion of the exhaust, the FRC is particularly interesting as a candidate to burn aneutronic fuels.
The FRC magnetic configuration has an ideal geometry for a future fusion propulsion utilizing D-3He fuel (see Fig. 1.11). As a matter of fact the null field region and the high beta mean low synchrotron radiation and moderate field requirements even at high plasma temperatures. Furthermore the linear geometry and the unimpeded outflow are natural for obtaining direct energy conversion .
Fig. 1.11. Idealized fusion propulsion utilizing D-3He fuel .
1.4. Spherical Tori
The spherical torus (A=R/a<2) was proposed in 1986 by M. Peng and D.J. Strickler (Oak Ridge National Laboratory, USA) , as a modification of the conventional tokamak, from which it differs because of a much smaller aspect ratio, see Fig. 1.12.
Fig. 1.12. Difference in aspect ratio between an ST and a conventional tokamak.
Peng and Strickler pointed out certain advantages connected with low A, concerning for examples the high value of the ratio between thermal energy and magnetic energy, b=2m0<p>Vol/B2 that can be achieved. No dedicated experiment along this line was built until the early '90. The first explorations of low aspect ratio configuration were made by modifying spheromak experiments with the addition of a central rod, carrying a current Itf, to produce a toroidal magnetic field. The objective was to control the tilting instability of the spheromak. The main results of this work was that a tokamak configuration could exist down to A=1.1. HSE (Heidelberg Spheromak Experiment) , Rotamak in Australia , SPHEX in UK , FBX II in Japan , were devoted to these experiments between 1987 and 1991, see Fig. 1.13. The drawbacks of the plasmas produced in these devices were the low temperature obtained (Te<50 eV) and the short pulse duration (tpulse<2 ms), which prevented a strong assessment on the feasibility and advantages of these configurations.
Fig. 1.13. Schemes of the first-generation ST experiments (1987-1991).
START at Culham [31, 32] began operation in 1991. Plasma currents up to Ip=250 kA were obtained by first inducing two current carrying plasma tori with large major radius. Thereafter they were merged and compressed down to A~1.25. A pulse length of ~40 ms, (extended by the addition of a compact central solenoid), allowed the attainment of hot (Te~500 eV) and dense (ne>1020 m-3) plasmas. Thus some characteristics of spherical tori could be, for the first time, compared with theoretical expectations with some confidence. CDX-U at Princeton [33, 34] and HIT at Seattle  completed the series of the experiments that produced spherical tori of sufficient duration to enable plasma properties to be evaluated. HIT is particularly relevant for PROTO-SPHERA (see Section 2.7), since it was devoted to the study of helicity injection to drive the plasma current. Indeed up to Ip=200 kA has been driven by this mechanism for ~20 ms, with a good power efficiency. Fig. 1.14 shows the schemes of the magnetic equilibria achieved in START, CDX-U and HIT.
Fig. 1.14. Schemes of the second-generation ST experiments (1991-1994).
After October 1995 START was modified by the installation of additional poloidal field coils , allowing for the obtainment of a divertor plasma, see Fig. 1.15.
Fig. 1.15. Visible light image of an X-point plasma in START.
Using a 1 MW neutral beam injector (NBI) START has reached an average toroidal beta value bT0=2m0<p>Vol/BT02~40% , where BT0 is the vacuum toroidal field on the magnetic axis. The very encouraging results of START have provided a good basis for building new ST experiments, capable of carrying currents in the MA range; the two main ST of this category now in operation are MAST and NSTX. In Fig. 1.16 the parameters of the two devices are compared.
Operation: 1999 1999
Major radius: R = 0.7 m R = 0.85 m
Minor radius: a = 0.5 m a = 0.68 m
Elongation: k ≤ 2 k ≥ 2
Toroidal field: BT ≤ 0.6 T BT ≥ 0.3 T
ST plasma current: Ip ≤ 2 MA Ip ≤ 2 MA
Plasma duration: tPlasma ≤ 2 s tPlasma ≤ 5 s
Additional power: Padd = 6.5 MW Padd = 12 MW
Fig. 1.16. Schemes and parameters of MAST and NSTX (third-generation ST).
MAST (Mega Amp Spherical Tokamak), has been built by the Culham Laboratory [37, 38], which is a scaled up version of START, has produced its first plasma in 1999 (Fig. 1.17.a) and 1 MA plasma in May 2000. MAST is endowed with a central solenoid, which can sustain an ohmic spherical tokamak. Its main aim is to extend the experimental results of START to the 1÷2 MA range of plasma current. Two broad objectives are: 1) to make a significant contribution to the understanding of tokamak physics (confinement scaling, plasma exhaust, MHD stability, etc.); 2) to test the spherical tokamak concept, in order to provide a database for a possible future Material Test Facility. In more detail, most interest is devoted to the study of exhaust in divertor configuration at high density; to the exploration of energy confinement properties, essentially to the dependence on the aspect ratio; to the study of the characteristics of the H-mode in these configurations; to the exploration of the operational limits (plasma density ne, average toroidal beta value bT0, safety factor at the edge qy or at 95% of the poloidal flux q95) and to the MHD stability properties; to the investigation of the efficiency of current drive systems (neutral beams). Additional heating is planned after the ohmic phase, and it is based on neutral beam injection (~5 MW) and on 60 GHz electron cyclotron heating (~1.5 MW).
Fig. 1.17. a) Visible light image of one of the first Ip=300 kA discharges in MAST.
b) Fast camera image of an Ip=500 kA ohmic discharge in NSTX.
c) Same for helicity injection startup at Ip=50 kA in NSTX.
NSTX (National Spherical Tokamak Experiment) is a very low aspect ratio (A~1.25) device built as a national facility by the Princeton Laboratory [39, 40] and has produced its first 1 MA plasma in December 1999 (Fig. 1.17.b and 1.17.c). Also NSTX is endowed with a central solenoid. The plasma current is in the same range as that of MAST, while its objectives are different. Apart from exploring confinement scaling and q limits, the main goal is to achieve and explore reactor relevant ST regimes, characterized by low collisionality, high b, high bootstrap current fraction fBS [41, 42, 43], at fully relaxed current density profiles. Thus from the beginning NSTX is designed having in mind the additional heating and current drive systems, with a capability of magnet pulse up to 5 seconds. It has to be noted that NSTX relies on helicity injection for plasma start-up and for edge current drive and has a conducting shell to help the plasma formation and the high beta stability.
The aspect ratio (A=R/a) of the spherical torus (ST) plasma approaches unity (A=1.1÷1.6 typically), compared to A=2.5÷5.0 for the conventional tokamak. Its magnetic surfaces combine a short field line of bad curvature and high pitch angle toward the outboard plasma edge with a long field line of good curvature and low pitch angle toward the inboard plasma edge (see Fig. 1.18).
Another feature of ST is that the geodesic curvature of the lines of force is almost zero on the inboard of the torus; at high b value a magnetic well (local minimum |B|) appears on the outboard of the torus. Therefore the small value of the banana width (quasi omnigeneity) and the time-averaged concentration of the trapped particle orbits in favorable curvature region  could limit the micro-instabilities related to trapped particles.
Fig. 1.18. Magnetic lines of force in a conventional tokamak and in an ST.
As the aspect ratio is reduced, for instance from A = 2.5 to A = 1.2, the elongation increases naturally, i.e. maintaining a uniform vertical field with a null field index (see Fig. 1.19). ST plasmas with cross-sections elongated up to k = 3 can have intrinsic vertical stability.
Fig. 1.19. Reducing the aspect ratio, the elongation of free boundary equilibria increases.
In an ST the poloidal field Bpol is comparable to the toroidal field BT, whereas in a tokamak Bpol«BT. As a result, the ST uses a modest applied toroidal field (TF), but has large values of the normalized current Ip/aBT and of the ratio between the plasma current and the toroidal field current Ip/Itf. High toroidal plasma current Ip can be driven with low toroidal field BT and with very simple windings, compared to conventional tokamaks (see Fig. 1.20). This corresponds to a high ohmic power density and to operations at high plasma density.
Fig. 1.20. Comparison of TF and PF field coils between ASDEX-UP and START.
The dominance of good field line curvature leads to magnetohydrodynamic (MHD) stability at high plasma pressure, giving the potential for order-unity average toroidal beta, bT0=2m0<p>Vol/BT02, and of order-10 normalized beta, bN=bT0aBT/Ip [%, m, T/MA]. The high beta and the magnetic configuration combine to widen the parameter domain for magnetic fusion plasmas.
Ideal MHD calculations for highly elongated ST (k=2.5÷3) show first stability beta limits in excess of bT0~50% (bN~4), in absence of any stabilizing wall near the plasma, and second stable regime at bT0=100% (bN~8), with a conducting shell at rshell/a<1.2 . An advantage of spherical tori is the ability to achieve second stability with monotone q-profiles, thus avoiding instabilities associated with low shear (infernal modes) or inverted q-profiles (double tearing).
The total bT0 value in magnetic confinement expresses how a plasma can be well confined in an apparatus of reduced size and cost (the ideal aim being bT0~1). The poloidal beta value bpol=2m0<p>Vol/Bpol2 marks instead the distance of the configuration from a force-free state (=0). In an ST, as Bpol~BT, bT0~1 means only bpol~1; therefore a high bT0 (40%) plasma in an ST is much nearer to a force-free configuration than a low bT0 (4%) plasma in a conventional tokamak. The high physical beta (referred to the magnetic energy contained in the plasma) of ST is even more significant as it still corresponds to a high engineering beta (referred to the total magnetic energy contained in the assembly). This statement is not true for other high beta configurations such as the high aspect ratio advanced tokamaks (AT).
The START experiment has demonstrated the high b-potential of ST achieving a toroidal volume average bT0=2m0<p>Vol/BT02=40%, with a peak toroidal bT0(0)=2m0p(0)/BT02=70%, and a normalized plasma beta bN~6 (see Fig. 1.21).
Fig. 1.21. Volume average bT0 versus Ip/aBT0 for START, compared to conventional
tokamaks; all the bT0 value in excess of 6% are due only to DIII-D, which
is the lowest aspect ratio conventional tokamak (A=2.5) .
The high bT0 values of START are even more noteworthy as they could be sustained for several confinement times in low q95~3 discharges, characterized by a ratio Ip/Itf~1 and furthermore in presence of an improved confinement regime (see Fig. 1.22) .
Fig. 1.22. Traces of a high-b shot (#35533) of START .
START has exhibited relatively high energy confinement times and density limits. H-mode like signatures have been observed in NBI heated double X-point discharges. Pellet injection has greatly extended the operating space and the density limits beyond the Greenwald limit (Fig. 1.23) .
Fig. 1.23. Confinement in START with NBI .
From ST equilibrium calculations a very low shear is expected on the central part of the plasma cross-section, and a very high shear occurs at the plasma edge. This could lead to stabilization of MHD and micro-instabilities and eventually to a favorable energy confinement. As a matter of fact on START, no current disruption was observed for A<1.8 [48, 49], at least before the installation of the X-point coils. Sawteeth and internal reconnection were still present, but did not destroy the plasma.
All the advanced features associated with ST (stabilization of MHD and micro-instabilities) are already accessible at low beta and do not depend upon the uncertain achievement of a high beta, as it is the case for the high aspect ratio advanced tokamak (see Fig. 1.24).
CONVENTIONAL ADVANCED TOKAMAK
SPHERICAL TORUS ADVANCED TOKAMAK
Fig. 1.24. Schematic comparison of the advanced tokamak concept, following the more conventional high aspect ratio or the more innovative spherical torus approach.
So it is conceivable that the two large ST experiments now operating, MAST and NSTX will successfully address a number of advanced tokamak issues.
The general scientific objectives that can be studied in the ST experiments are:
Achievement of bT0 =1 at aspect ratio A®1.
Test of resistive and neo-classical MHD at A®1.
Stabilization of micro-instabilities.
Well aligned bootstrap current (the bootstrap current fraction fBS can approach 100% for elongation k®3).
Overlap of ST, spheromak and FRC.
The advantages of the ST in the path toward the development of an economically attractive fusion power source can be so summarized:
High Q path to reactor cheaper and faster than with conventional tokamaks.
Possibility of trying different blanket concepts and nuclear engineering components on a number of cheap experimental power sources (as it has been for fission reactors).
Reduced waste volume.
1.5. The Ultra Low Aspect Ratio Torus (ULART)
The usual configuration of an ST is connected with a slim central rod that carries the current necessary to create the toroidal magnetic field. Thus a central solenoid coil can store only a small inductive flux. Due to the low inductance of the ST, this flux can be sufficient to bring the current to its nominal value, but then a non-inductive current drive system is needed to maintain the current during the flat top. This restriction is even more severe if one aims at A<1.3, where many advantages are obtained, according to the theory. Many systems of non-inductive current drive seem of difficult or impossible applicability [50, 51]. In the RF range of frequencies, only fast wave current drive has up to now been considered  for proposed experiments. Neutral beams [53, 54, 55] and helicity injection [56, 57] are the systems most considered for implementation.
The main problems of the ST to be solved in the path toward the development of an economically attractive fusion power source can be so summarized:
Achievement of reliable start-up techniques in absence of an ohmic transformer.
Demonstration of reliable current drive (based either upon bootstrap and non-inductive methods) on ST.
Choice of optimal aspect ratio.
Feasibility of single turn central rod for the toroidal field coil in order to achieve an easy maintenance and substitution.
The limits to aspect ratio have been explored in the TS-3 experiment [58, 59] at the University of Tokyo, used as a spherical torus. Record of low aspect ratio A=1.1÷1.2, with ratio of the toroidal field current and plasma current as low as Itf/Ip=0.20, have been achieved (see Fig. 1.25).
Fig. 1.25. Limit to Itf/Ip at low aspect ratio in TS-3 .
Along with the results of TS-3, other reasons push towards the Ultra Low Aspect Ratio Tokamaks (ULART, A<1.3). One of the main problems in designing an ST reactor is the central conductor that creates the toroidal magnetic field. It cannot be shielded, is bombarded by neutrons and so cannot be built by superconducting materials and can involve too large an energy dissipation with respect to the produced fusion power. The only way to avoid an excessive dissipation in the central conductor is to go down in aspect ratio until A<1.3 . This is allowed by the ratio Itf/Ip which, for aspect ratios A<2, is well described by the Katsurai formula :
Itf/Ip=p2qy(A-1)2/(2k2) (see Fig. 1.26).
Fig. 1.26. Behavior of magnetic field lines in an ULART.
This relation is calculated from the behavior of the lines of force in the inboard region, near the central rod:
BT=m0Itf /2prtf , Bpol=m0Ip/4b , 1/qy= 2prtf Bpol/2bBT , A=1+rtf /a.
Fixing the value of the safety factor to qy=3 one can see that, whereas in an ST with aspect ratio A=1.5 and elongation k=2.0, Itf/Ip=1.2, in an ULART with aspect ratio A=1.2 and k=3, Itf/Ip=0.1.
Another limit to A is given by an analytic treatment of the rigid tilt instability, which is calculated  as:
with CL (of order of unity), determined by the vertical field BZ=CLm0Ip/(4pRp), and n* (very near to zero), determined by n*=-(R/BZ)dBZ/dR.
The two limits (see Fig. 1.27), with CL = 0.645 and n* = 0.07 are:
Itf/Ip ≥ 4.93 qy (A-1)2/ k2 for the qy limit
Itf/Ip ≥ 1.94 A1/2 (A-1) / k3/2 for the tilt limit
With k =2.3 and qy = 3 the two limits coincide at A=1.2; at A>1.2 the qy sets the current limit but at A<1.2 it is the tilt which limits the plasma current.
Fig. 1.27. Itf/Ip limit at low aspect ratio with k=2.3 and qy=3.
The ULART configuration however does not leave enough space for a central ohmic transformer and so requires non-inductive current drive methods. Furthermore the ULART configuration does not solve the problem of the neutron damaging of the central conductor in a D-T reactor.