High-beta Equilibria Research Articles

Optimized profiles for improved confinement and stability in the DIII-D tokamak

Simultaneous achievement of high energy confinement, tau E, and high plasma beta, beta , leads to an economically attractive compact tokamak fusion reactor. High confinement enhancement, H= tau E/ tau E-ITER89P=4, and high normalized beta beta N beta /(I/aB)=6%-m-T/MA have been obtained in DIII-D experimental discharges. These improved confinement and/or improved stability limits are observed in several DIII-D high performance operational regimes: VH-mode, high li H-mode, second stable core, and high beta poloidal. We have identified several important features of the improved performance in these discharges: details of the plasma shape, toroidal rotation or E*B flow profile, q profile and current density profile, and pressure profile. From our improved physics understanding of these enhanced performance regimes, we have developed operational scenarios which maintain the essential features of the improved confinement and which increase the stability limits using localized current profile control. The stability limit is increased by modifying the interior safety factor profile to be nonmonotonic with high central q, while maintaining the edge current density consistent with the improved transport regimes and the high edge bootstrap current. We have calculated high beta equilibria with beta N=6.5, stable to ideal n=1 kinks and stable to ideal ballooning modes. The safety factor at the 95% flux surface is 6, the central q value is 3.9 and the minimum in q is 2.6.

Magnetic surface breaking in three-dimensional l=2 heliotron/torsatron equilibria

Magnetic surface breaking due to finite pressure effect is analyzed for three-dimensional magnetohydrodynamic equilibria of l=2 heliotron/torsatron configuration. Characteristics of magnetic island formation are numerically studied in detail. Main results are as follows: (i) There exists a clear-cut boundary, which practically determines the position of the outermost closed surface. The boundary position shifts as beta increases, though not affected by the external axisymmetric fields. (ii) The clear-cut boundary is maintained for a higher beta plasma as the aspect ratio of the configuration increases. For a low aspect ratio configuration, however, high beta equilibria with clear surfaces can be obtained by controlling the external poloidal field, especially the vertical field. (iii) The external l=1 field is particularly viable in controlling the breaking of magnetic surfaces.

Kinetic high-beta equilibria in a plasma slab

Self-consistent, high-beta kinetic equilibria are derived from the Vlasov equation for a plasma slab in a nonzero magnetic field with a density gradient and a gravitational field. It is shown that a magnetic field inhomogeneity and an electric field arise as necessary consequences of the high-beta state. Hence the corresponding equilibrium forces should not be neglected in calculations of individual particle motion. Explicit expressions are derived for equilibrium quantities, useful, e.g., for large Larmor radius (LLR) studies of the Rayleigh–Taylor instability.

An analytic solution of high-beta equilibrium in a large aspect ratio tokamak

An analytic solution of the high-beta (εβ̄p∼βq2/ε≫1) equilibrium of a large aspect ratio tokamak is presented. Two arbitrary flux functions, the pressure profile p(ψ) and the safety factor profile q(ψ), specify the equilibrium. The solution splits into two asymptotic regions: the core region where ψ is a function of the major radius alone and a narrow boundary layer region adjoining the conducting wall. The solutions in the two regions are asymptotically matched to each other. For monotonic pressure profiles, the Shafranov shift is equal to the minor radius. For beta much bigger than 1, the solution contains a region (in place of the magnetic axis) of zero magnetic field and constant pressure. At high beta the quantity βI, which is essentially proportional to the pressure over the total current squared, is largely independent of pressure. The important ramifications of limited βI for high-beta reactors are discussed. Generalizations to shaped cross sections and hollow pressure profiles are outlined. The problem of equilibrium reconstruction in the high-beta regime is also considered.

High poloidal beta equilibria in the Tokamak Fusion Test Reactor limited by a natural inboard poloidal field null

Recent operation of the Tokamak Fusion Test Reactor (TFTR) [Plasma Phys. Controlled Nucl. Fusion Research 1, 51 (1986)] has produced plasma equilibria with values of Λ≡βp eq+li/2 as large as 7, εβp dia≡2μ0ε〈p⊥〉/〈〈Bp〉〉2 as large as 1.6, and Troyon normalized diamagnetic beta [Plasma Phys. Controlled Fusion 26, 209 (1984); Phys. Lett. 110A, 29 (1985)], βNdia≡108〈βt⊥〉aB0/Ip as large as 4.7. When εβp dia≳1.25, a separatrix entered the vacuum chamber, producing a naturally diverted discharge that was sustained for many energy confinement times, τE. The largest values of εβp and plasma stored energy were obtained when the plasma current was ramped down prior to neutral beam injection. The measured peak ion and electron temperatures were as large as 24 and 8.5 keV, respectively. Plasma stored energy in excess of 2.5 MJ and τE greater than 130 msec were obtained. Confinement times of greater than 3 times that expected from L-mode predictions have been achieved. The fusion power gain QDD reached a value of 1.3×10−3 in a discharge with Ip=1 MA and εβp dia=0.85. A large, sustained negative loop voltage during the steady-state portion of the discharge indicates that a substantial noninductive component of Ip exists in these plasmas. Transport code analysis indicates that the bootstrap current constitutes up to 65% of Ip. Magnetohydrodynamic (MHD) ballooning stability analysis shows that, while these plasmas are near, or at the βp limit, the pressure gradient in the plasma core is in the first region of stability to high-n modes.

An analytic large aspect ratio, high beta equilibrium

Understanding the magnetohydrodynamic equilibrium and stability properties of Tokamaks with extreme values of plasma beta is a problem of eternal interest. Numerical work is adequate for elucidating the practical consequences, but it is useful, when taking a parameter to its limit, to have analytic or semi-analytic models. Here an axisymmetric, toroidal, high-beta equilibrium model is constructed that is superior to those used in the past. It demonstrates the nature of the limit of very large beta, and should be useful as a starting point for stability studies.

Stability analysis for the Big Dee upgrade of the Doublet III tokamak

Ideal magnetohydrodynamic stability analysis has been carried out for configurations expected in the Big Dee tokamak, an upgrade of the Doublet III tokamak into a non-circular cross-section device which began operation early in 1986. The results of this analysis support theoretical predictions as follows: Since the maximum value of beta stable to ballooning and Mercier modes, which we denote βc, increases with inverse aspect ratio, elongation and triangularity, the Big Dee is particularly suited to obtain high values of βc and there exist high βc Big Dee equilibria for large variations in all relevant plasma parameters. The beta limits for the Big Dee are consistent with established theory as summarized in present scaling laws. High beta Big Dee equilibria are continuously accessible when approached through changes in all relevant input parameters and are structurally stable with respect to variations of input plasma parameters. Big Dee beta limits have a smooth dependence on plasma parameters such as βp and elongation. These calculations indicate that in the actual running of the device the Big Dee high beta equilibria should be smoothly accessible. Theory predicts that the limiting plasma parameters, such as beta, total plasma current and plasma pressure, which can be obtained within the operating limits of the Big Dee are reactor relevant. Thus the Big Dee should be able to use its favourable ideal MHD scaling and controlled plasma shaping to attain reactor relevant parameters in a moderate sized device.

Tokamak Power Systems Studies at ANL

A number of advances in plasma physics and engineering promise to greatly improve the reactor prospects of tokamaks. The following features, in particular, are examined: (a) large aspect ratio (A approx. = 6), which may ease maintenance; (b) high beta (..beta.. greater than or equal to 0.20) without indentation, which brings the maximum toroidal field down to about 7 T; (c) low toroidal current (I approx. = 5MA), which reduces the cost of the current drive and equilibrium field system; and (d) steady state operation with current density control via fast and slow wave current drive. The key to high beta operation with low toroidal current lies in utilizing second stability regime equilibria with the required current distributions produced by an appropriate selection of wave driver frequencies and power spectra. The ray tracing and current drive calculation is self-consistent with the actual magnetic fields produced in the plasma. In addition to matching desirable high-beta equilibria, this method is capable of producing a large variety of new equilibria, many of which look attractive. The impurity control activities in TPSS have emphasized the self-pumping concept as applied to using the entire first wall or ''slot'' limiters. The blanket design effort has emphasizedmore » liquid metal and Flibe concepts. The reference concept is a liquid lithium/vanadium, self-cooled configuration. Overall, there exists a number of major design improvements which will substantially improve the attractiveness of tokamak reactors.« less

Zero-current high-beta stellarator equilibria with rotational transform profile control

Zero-current high-beta equilibria for a stellarator device are calculated using the averaging method. It is found that, by shaping the vertical field, the rotational transform can be controlled in an approximate way as beta is increased. At the same time, the Pfirsch-Schlüter currents are reduced – with no modification of the magnetic well. This permits access to the high-beta regime with more favourable rotational transform profiles. Results are presented for the Advanced Toroidal Facility (ATF) device.

Use of the stellarator expansion to investigate plasma equilibrium in stellarators

A numerical code utilizing a large-aspect-ratio, small -helical-distortion expansion is developed and used to investigate the effect of plasma currents on stellarator equilibrium. Application is made to several existing and proposed stellarators. The modification of the equilibrium properties with pressure is strongly dependent on the vacuum-field configuration. The modular stellarator model which is investigated gives somewhat discouraging results because a large rotational transform, and hence large coil deformation, is needed to achieve high-beta equilibria.

Plasma Equilibrium and Field Diffusion during Current Rise Phase of STP-2 Experiments

The plasma with maximum poloidal beta value β p (the ratio of the plasma pressure to the magnetic pressure of the toroidal plasma current) of 3.0 has been obtained in STP-2 by compression together with joule heating. In this case of β p =3.0 the plasma interacts strongly with wall. The maximum β p value without strong wall contacts was about 1.3. We observed that force-free current is formed in the periphery of the plasma and the penetration of the poloidal magnetic field is much faster than the classical one. We have performed a numerical simulation using a 2- D MHD pinch code “ TOPICS ” in order to understand this high-beta plasma equilibrium having β p =3.0 and the mechanism of fast field penetration observed in STP-2 plasmas. It is demonstrated that the fast field penetration and the observed electron temperatures can be explained by introducing the influx of neutral particles and the ion acoustic type anomalous resistivity.

Stabilité magnétohydrodynamique des modes ballooning dans un Tokamak

We implanted in the ERATO spectral code [1] the stability criterion for ballooning modes (toroidal wave numbern→∞) [2]. The study of ballooning waves located around a given magnetic surface reduces itself to the investigation of the sign of the potential energy on this surface; this method avoids the determination of eigenvalues which is expensive in computing time. The stability criterion obtained by this way has been applied to a high-beta equilibria family proposed for JET. The stability limit found is more constraining than the limit for low-n internal modes.

MHD Stability for a Class of Tokamak Equilibria with Fixed Boundaries

The stability behavior with respect to internal modes is discussed for a class of tokamak equilibria with non-circular cross-sections and essentially flat current profiles. The stability analysis is done by computer both symbolically and numerically with the help of a normal mode code, which extremizes the Lagrangian of the system . It is found that the stability limit agrees well with that of the Mercier criterion. There are stable high-beta equilibria in this model.