ALBERT

Notes: The previously developed single-gap kinetic theory for toroidicity-induced Alfvén eigenmodes (TAE) is extended and applied to Doublet III-D [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)] experimental data. It is found that the theory gives reasonable agreement with the data when an appropriate magnetohydrodynamic boundary condition is accounted for. As is shown, this boundary condition is equivalent to an appropriate real frequency shift relative to the continuum gap. The correct eigenfunction near the gap, and thus the correct damping, is obtained by using the gap structure calculated from an equilibrium reconstruction that includes low aspect ratio, noncircularity, and finite beta effects, combined with an experimentally measured frequency. In the considered experimental cases, the damping is well into the nonideal regime.

Notes: Stability at high beta (the ratio of plasma pressure to magnetic field pressure) is an important requirement for a compact, economically attractive fusion reactor. It is also important in present large tokamak experiments, where the best performance is now often limited by instabilities rather than by energy transport. The past decade has seen major advances in our understanding of the stability of high beta tokamak plasmas, as well as in the achievement of high values of beta. Ideal magnetohydrodynamic (MHD) theory has been remarkably successful in predicting the stability limits, and the scaling of maximum stable beta with the normalized plasma current predicted by Troyon and others has been confirmed in many experiments, yielding a limit βmax≈3.5 (%-m-T/MA) I/aB (where I is the plasma current, a is the minor radius, and B is the toroidal field). The instabilities which are predicted to limit beta have been observed experimentally, in good agreement with theoretical predictions, including long-wavelength kink modes and short-wavelength ballooning instabilities. Advances in understanding of tokamak stability have opened several paths to higher values of beta. The use of strong discharge shaping, approaching the limits of axisymmetric stability, has allowed beta values as high as 12% to be reached in agreement with Troyon scaling. Recent experimental results and ideal MHD modeling have shown that the beta limit depends on the form of the pressure and current density profiles, and modification of the current density to create a centrally peaked profile has allowed beta values up to 6I/aB to be achieved experimentally.Recent experiments have also begun to explore both local and global access to the predicted second stable regime for ballooning modes, with the potential for very high values of β/(I/aB). Preliminary experimental investigations of wall stabilization and radio-frequency (RF) current profile control hold the promise of further improvements in beta through passive and active control of instabilities. The developing understanding of high beta stability and the application of this understanding to present experiments and future fusion devices hold the potential for production of stable, steady state plasmas at high beta with good confinement.

Notes: Intense fast-ion populations created by neutral-beam injection into a tokamak can destabilize toroidicity-induced Alfvén eigenmodes (TAE modes) or internal kink modes. Experimentally, these modes stabilize when fast ions are ejected from the plasma, producing a cycle of relaxation oscillations about the marginal stability point. A pair of coupled differential equations describes this cycle. This simple theoretical formalism successfully describes the cycles observed during TAE experiments in DIII-D [Plasma Physics Controlled Nuclear Fusion Research, 1986 (International Atomic Energy Agency, Vienna, 1987), Vol. 1, p. 159].

Notes: The effects of current profile on the ideal ballooning mode for circular and shaped poloidal cross-section plasmas in tokamaks are studied analytically and numerically. The results show that for moderately shaped plasmas the critical normalized beta, βNC, against the ballooning mode increases approximately linearly with the plasma internal inductance li. As the plasma becomes more strongly shaped, this dependence on li becomes weaker, and for a divertor plasma βNC shows a very weak dependence on li for the range of moderate li values considered.

Notes: Accurate equilibrium reconstruction and detailed stability analysis of a strongly shaped, double-null, βT=11% discharge shows that the plasma core is in the second stable regime to ideal ballooning modes. The equilibrium reconstruction using all the available data (coil currents, poloidal magnetic loops, motional Stark effect data, the kinetic pressure profile, the magnetic axis location, and the location of the two q=1 surfaces) shows a region of negative magnetic shear near the magnetic axis, an outer positive shear region, and a low shear region connecting the two. The inner negative shear region allows a large positive shear region near the boundary, even at low q (q95=2.6), permitting a large outer region pressure gradient to be first regime stable. The inner region is in the second stable regime, consistent with the observed axial beta [βT(0)=44%]. In the low shear region p' vanishes, consistent with Mercier stability. This is one way to extend the ballooning limit in shaped plasmas while maintaining stability against external kinks. The n=1 analysis shows that the plasma is unstable to an ideal internal mode, consistent with the experimental observations of a saturated internal m/n=1/1 mode. The core plasma pressure, not being limited by ballooning stability, appears to be reaching a local equilibrium limit at the magnetic axis.

Notes: Improvement in both the energy confinement time and the achievable value of normalized beta is obtained by modifying the current density profile from the relatively broad shape obtained in standard tokamak discharges to a more peaked shape. The peaked current profile is produced with either a rapid negative ramp in the total plasma current or a rapid increase in the discharge elongation. Discharges have been obtained with βN=β/(I/aB)=6% mT/MA simultaneously with total energy confinement time two times the value predicted by L-mode scaling relations. Up to a factor of 1.8 improvement in the normalized thermal energy confinement time, τth/Ip, has been obtained in both L-mode and H-mode discharges. It is shown that the increase in confinement can be attributed to a local decrease in the thermal diffusivity that is correlated with a local increase in the poloidal magnetic field and the magnetic shear.

Notes: The stability of the n=1 ideal kink in DIII-D-like [Fusion Technol. 8, 441 (1985)] configurations is found to depend critically on the details of the current density and pressure profiles. The maximum stable normalized β, βN=β/(I/aB) (I in MA, a in meters, and B0 in tesla), is found to increase dramatically as the peak in the pressure gradient is shifted from the central region toward the plasma boundary. Further, for peaked pressure profiles, the β limit is insensitive to the internal inductance, li, whereas for broad pressure profiles, the β limit depends more strongly on li and can increase with li almost up to the point where the q profile becomes hollow. With broad pressure profiles and a conducting wall at 1.5 minor plasma radii, kink-stable βN values of 5.8−significantly above both the Troyon limit and the ballooning limit−have been found.

Notes: A regime of very high confinement (VH mode) has recently been observed in DIII-D with global energy confinement times up to a factor of 3.5 above the ITER89-P L-mode scaling [Nucl. Fusion 30, 1999 (1990)] and 1.5 to 2 times greater than the DIII-D/JET edge-localized-mode-free H-mode scaling relation [Nucl. Fusion 31, 73 (1991)]. These discharges were obtained after boronization in DIII-D and are characterized by low radiated power and Zeff, increasing confinement time during the VH phase of the discharge and low Ohmic target density. The low radiated power and Zeff are a consequence of the boronization. During the VH phase these discharges exhibit an inward shift in the region of highest electric field shear and a large calculated edge bootstrap current. The outer region (ρ(approximately-greater-than)0.85) is calculated to be in the second stable regime to ideal ballooning modes.

Notes: A variational ideal magnetohydrodynamic stability code is used to compute the toroidicity-induced Alfvén eigenmode (TAE) for toroidal mode number n=1 in full toroidal geometry with circular cross section. At finite aspect ratio, toroidicity also couples poloidal mode numbers m with m+2 to produce a higher frequency gap in the shear Alfvén spectrum, which is associated with a new, second-order TAE mode. It is also found that toroidal coupling between the TAE modes and continuum branches induces splitting of the TAE modes into two or more global modes at slightly different frequencies. Both the new modes and splitting of the TAE modes have important consequences for the identification of TAE modes in experiments.

Notes: A theoretical and experimental evaluation of axisymmetric stability and axisymmetric control has led to a modification of the vertical position control in the DIII-D tokamak, which now allows operation to within a few percent of the ideal magnetohydrodynamic (MHD) n=0 limit. It is found that the onset the departure from rigid shift behavior in D-shaped plasmas limits plasma elongation to 2.5 in DIII-D. The possibility of avoiding the vertical instability in future tokamaks with highly elongated plasmas is discussed. Recent experiments have focused on utilizing this capability for axisymmetric control to construct plasma shapes optimized to increase the achievable beta. Operation near the axisymmetric stability limit allows an increase in the achieved normalized current Ip/aBT, where Ip is the total plasma current, a is the minor radius, and BT is the toroidal field. Based on stability calculations, an equilibrium was developed to achieve marginal stability simultaneously to axisymmetric, kink, and ballooning instabilities. In the experiment, the shape was altered to higher elongation during the high-beta phase as the current profile broadened. A record high beta for DIII-D of 11% was achieved. The high-beta phase of the discharge lasted 40 msec, approximately one confinement time.