Beta Limit Research Articles

Nonlinear subcritical magnetohydrodynamic beta limit

Published gyrokinetic simulations have had difficulty operating beyond about half the ideal magnetohydrodynamic (MHD) critical beta limit with stationary and low transport levels in some well-established reference cases. Here it is demonstrated that this limitation is unlikely due to numerical instability, but rather appears to be a nonlinear subcritical MHD beta limit [R. E. Waltz, Phys. Rev. Lett. 55, 1098 (1985)] induced by the locally enhanced pressure gradients from the diamagnetic component of the nonlinearly driven (zero frequency) zonal flows. Strong evidence that the zonal flow corrugated pressure gradient profiles can act as a MHD-like beta limit unstable secondary equilibrium is provided. It is shown that the addition of sufficient E×B shear or operation closer to drift wave instability threshold, thereby reducing the high-n drift wave turbulence nonlinear pumping of the zonal flows, can allow the normal high-n ideal MHD beta limit to be reached with low transport levels. Example gyrokinetic si...

Demonstration of ITER operational scenarios on DIII-D**Presented at the 22nd IAEA Fusion Energy Conf. (Geneva, Switzerland, 13–18 October 2008) http://www-pub.iaea.org/MTCD/Meetings/fec2008pp.asp

The DIII-D programme has recently initiated an effort to provide suitably scaled experimental evaluations of four primary ITER operational scenarios. New and unique features of this work are that the plasmas incorporate essential features of the ITER scenarios and anticipated operating characteristics; e.g. the plasma cross-section, aspect ratio and value of I/aB of the DIII-D discharges match the ITER design, with size reduced by a factor of 3.7. Key aspects of all four scenarios, such as target values for βN and H98, have been replicated successfully on DIII-D, providing an improved and unified physics basis for transport and stability modelling, as well as for performance extrapolation to ITER. In all four scenarios, normalized performance equals or closely approaches that required to realize the physics and technology goals of ITER, and projections of the DIII-D discharges are consistent with ITER achieving its goals of ⩾400 MW of fusion power production and Q ⩾ 10. These studies also address many of the key physics issues related to the ITER design, including the L–H transition power threshold, the size of edge localized modes, pedestal parameter scaling, the impact of tearing modes on confinement and disruptivity, beta limits and the required capabilities of the plasma control system. An example of direct influence on the ITER design from this work is a modification of the physics requirements for the poloidal field coil set at 15 MA, based on observations that the inductance in the baseline scenario case evolves to a value that lies outside the original ITER specification.

Edge Stability and Pedestal Profile Sensitivity of Snowflake Diverted Equilibria in the TCV Tokamak

AbstractA second order null divertor (snowflake) has been successfully created and controlled in the TCV tokamak [1] (F. Piras et al., Plasma Phys. Control. Fusion, 2009). The results of ideal MHD edge stability computations show an enhancement of the edge stability properties of the snowflake equilibria compared to standard x‐point configurations [2] (S. Yu. Medvedev et al., 36th EPS Conference on Plasma Physics, 2009). However, a sensitivity study of the stability limits to variations of the pedestal profiles is essential for making conclusions about possibilities of ELM control in snowflake plasmas. Variations of the edge stability and beta limits for several types of snowflake equilibria, different values of triangularity and various pedestal profiles are investigated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Integrated Performance and Critical Issues for Steady-State Operation in JT-60U

This paper reports on the integrated performance achieved in JT-60U toward the steady-state operation foreseen in the ITER and DEMO reactors. Advanced tokamak plasmas with weak shear or reversed shear have been optimized to confront critical issues such as high-beta operation with high confinement, the compatibility of high-density operation with high confinement, and long sustainment with a high non-inductive current drive fraction. As a result, high-integrated performance was achieved in both plasma regimes. For example, high-confinement reversed shear plasmas with a high bootstrap current fraction exceeding the no-wall beta limit have been obtained in the reactor relevant q95∼5.3; high values of βN∼2.7, HH98y2∼1.7, ne/nGW∼0.87, and fBS∼0.9 are simultaneously achieved with a reversed q profile with qmin∼2.3.

Rotational stabilization of resistive wall modes in ITER advanced tokamak scenarios

Rotational stabilization of n=1 resistive wall modes in ITER advanced scenarios [K. Ikeda, Nucl. Fusion 47 (2007)] is investigated, where n is the toroidal mode number. In particular, we present numerical results for the ITER strongly reversed shear case, in comparison to the weakly reversed shear case. The rotation frequency is assumed to be modestly low. Our investigation employs the adaptive eigenfunction independent solution-kinetic (AEGIS-K) code [L. J. Zheng et al., “AEGIS-K code for linear kinetic analysis of toroidally axisymmetric plasma stability,” J. Comput. Phys. (to be published)], which provides a fully kinetic (nonhybrid) and self-consistent (nonperturbative) description. AEGIS-K includes wave-particle resonances, shear Alfvén continuum damping, trapped particle effects, and parallel electric effects, but not finite Larmor radius effects. In the case without rotation and kinetic effects included, we find that the strongly reversed shear configuration is more favorable for perfectly conducting wall stabilization of resistive wall modes, in that it has a higher conducting wall beta limit than the weakly reversed shear case. With sufficient rotation, the strongly reversed shear case can also achieve a higher beta limit for completely suppressing the resistive wall modes. However, the marginal rotation frequency required for complete resistive wall mode stabilization in the strongly reversed shear case is about twice as high as that required for the weakly reversed shear case.

“One size fits all” analytic solutions to the Grad–Shafranov equation

An extended analytic solution to the Grad–Shafranov equation using Solov’ev profiles is presented. The solution describes standard tokamaks, spherical tokamaks, spheromaks, and field reversed configurations. It allows arbitrary aspect ratio, elongation, and triangularity as well as a plasma surface that can be smooth or possess a double or single null divertor X-point. The solution can also be used to evaluate the equilibrium beta limit in a tokamak and spherical tokamak in which a separatrix moves onto the inner surface of the plasma.

Equilibrium and global MHD stability study of KSTAR high beta plasmas under passive and active mode control

The Korea Superconducting Tokamak Advanced Research, KSTAR, is designed to operate a steady-state, high beta plasma while retaining global magnetohydrodynamic (MHD) stability to establish the scientific and technological basis of an economically attractive fusion reactor. An equilibrium model is established for stability analysis of KSTAR. Reconstructions were performed for the experimental start-up scenario and experimental first plasma operation using the EFIT code. The VALEN code was used to determine the vacuum vessel current distribution. Theoretical high beta equilibria spanning the expected operational range are computed for various profiles including generic L-mode and DIII-D experimental H-mode pressure profiles. Ideal MHD stability calculations of toroidal mode number of unity using the DCON code shows a factor of 2 improvement in the wall-stabilized plasma beta limit at moderate to low plasma internal inductance. The planned stabilization system in KSTAR comprises passive stabilizing plates and actively cooled in-vessel control coils (IVCCs) designed for non-axisymmetric field error correction and stabilization of slow timescale MHD modes including resistive wall modes (RWMs). VALEN analysis using standard proportional gain shows that active stabilization near the ideal wall limit can be reached with feedback using the midplane segment of the IVCC. The RMS power required for control using both white noise and noise taken from NSTX active stabilization experiments is computed for beta near the ideal wall limit. Advanced state-space control algorithms yield a factor of 2 power reduction assuming white noise while remaining robust with respect to variations in plasma beta.

An improved method to evaluate the ideal no-wall beta limit from resonant field amplification measurements in JET

Modelling the low-n, low frequency resonant field amplification (RFA) effects for JET plasmas, using the MHD code MARS-F, offers explanations to one of the recent observations made in experiments, namely a mismatch between the measured RFA threshold and the predicted no-wall beta limit according to ideal MHD calculations. The mismatch is minimized by applying a new way of determining the RFA threshold, based on evaluating the logarithmic derivative of the RFA amplitude as a function of the normalized plasma pressure. This improved method is shown, at least in theory, to be robust in predicting the no-wall beta limit for JET plasmas.

Comprehensive control of resistive wall modes in DIII-D advanced tokamak plasmas

The resistive wall mode (RWM) and neoclassical tearing mode (NTM) have been simultaneously suppressed in the DIII-D for durations of over 2 s at beta values 20% above the no-wall limit with modest electron cyclotron current drive and very low plasma rotation. The achieved plasma rotation was significantly lower than reported previously. However, in this regime where stable operation is obtained, it is not unconditionally guaranteed. Various MHD activities, such as edge localized modes (ELMs) and fishbones, begin to couple to the RWM branch near the no-wall limit; feedback has been useful in improving the discharge stability to such perturbations. Simultaneous operation of slow dynamic error field correction and fast feedback suppressed the pile-up of ELM-induced RWM at a series of ELM events. This result implies that successful feedback operation requires not only direct feedback against unstable RWM but also careful control of MHD-induced RWM aftermath, which is the dynamical response to a small-uncorrected error field near the no-wall beta limit. These findings are extremely useful in defining the challenge of control of the RWM and NTM in the unexplored physics territory of burning plasmas in ITER.

Development of reversed shear plasmas with high bootstrap current fraction towards reactor relevant regime in JT-60U

This paper reports the recent development of reversed shear plasmas with a high bootstrap current fraction (fBS) towards the reactor relevant regime. The previous operation regime of high fBS plasmas is limited at q95 > 8 because of the low beta limit, whereas q95 = 5–6 is envisaged in the DEMO reactor. In the 2008 JT-60U experimental campaign, the high fBS plasma was emphasized in the lower q95 regime by developing the large volume configuration close to the conducting wall for wall stabilization. Thanks to the wall stabilization, high fBS plasmas exceeding the no-wall beta limit are obtained at reactor relevant q95 ∼ 5.3. Though the high fBS plasmas are terminated by destabilization of the resistive wall mode, a highly integrated performance is obtained. High values of HH98y2 ∼ 1.7, βN ∼ 2.7, fBS ∼ 0.92 and ne/nGW ∼ 0.87 are simultaneously achieved under the reactor relevant conditions of low momentum input and electron temperature nearly equal to ion temperature.

Preliminary Study of Ideal Operational MHD Beta Limit in HL-2A Tokamak Plasmas

Magnetohydrodynamic (MHD) n = 1 kink mode with n the toroidal mode number is studied and the operational beta limit, constrained by the mode, is calculated for the equilibrium of HL-2A by using the GATO code. Approximately the same beta limit is obtained for configurations with a value of the axial safety factor q0 both larger and less than 1. Without the stabilization of the conducting wall, the beta limit is found to be 0.821% corresponding to a normalized beta value of βcN = 2.56 for a typical HL-2A discharge with a plasma current Ip = 0.245 MA, and the scaling of βcN ∼constant is confirmed.

Analytic high-beta tokamak equilibria with poloidal-sonic flow

A reduced set of equations for high-beta tokamak equilibria with flow comparable to the poloidal sound velocity is solved analytically. The solution includes higher-order terms of asymptotic expansions in terms of the inverse aspect ratio and indicates the modification of the magnetic structure and the departure of the pressure surfaces from the magnetic surfaces by sub- or super-poloidal-sonic flows. The analytical representation for the shift of the magnetic axis from the geometrical axis (the Shafranov shift) and that of the pressure maximum and the equilibrium beta limit are also obtained. The Shafranov shift is enhanced by a slightly super-poloidal-sonic flow and it produces a forbidden region of equilibrium by the poloidal-sonic flow. The physical mechanism of the shift of the pressure maximum from the magnetic axis due to the poloidal-sonic flow is discussed in analogy to those of the geodesic acoustic mode and the slow magnetosonic wave.

Experimental studies of stability and beta limit in JET

The paper presents results on the use of resonant field amplification for experimental probing of stability and β-limits (β is the ratio of the plasma pressure to the magnetic field pressure) in JET. It is found that an externally applied helical magnetic field is strongly enhanced when the plasma exceeds the ideal no-wall stability limit or approaches proximity to other marginally stable (i.e. current-driven) modes. This effect is known as the resonant field amplification (RFA) and was used for the systematic probing of stability in different advanced regimes on JET. The application of this technique on JET is discussed in the paper and the results of the RFA measurements are presented and related to the observed limitations in β.

Beta limit due to m/n = 2/1 tearing mode onset in the DIII-D hybrid scenario

The performance (βN ≈ 3, q95 ≈ 4.4, fBS ≈ 0.5, H89 > 2) of hybrid scenario plasmas in DIII-D (Luxon 2002 Nucl. Fusion 42 614) is limited by m/n = 2/1 tearing modes. Unlike conventional plasmas (Hender et al 2004 Nucl. Fusion 44 798), the linear dependence scaling of the global beta for onset of the instability with normalized local ion gyroradius is modified as the n = 1 ideal kink beta limit is approached, suggesting that small island neoclassical tearing mode (NTM) threshold physics does not impose the dominant criterion for NTM stability. The hybrid scenario tends to go unstable just at or below the no wall n = 1 ideal kink beta limit of about 4ℓi. Experimentally 4ℓi decreases with beta as . Thus the ‘ceiling’ in beta due to coupling of tearing to the ideal kink comes down as beta is increased. Scaling of the tearing unstable beta that combines both NTM threshold physics and a pole in Δ′r due to coupling to the ideal kink is presented.

Formation, sustainment and characteristics of current hole plasmas in DIII-D discharges

Plasmas with zero or near-zero current density in a wide region of the core have been produced in DIII-D discharges. The discharges were obtained with early neutral beam and electron cyclotron heating (ECH) during the plasma current ramp. Wide current holes (CHs) with a near-zero field over about 23% of the plasma width and narrow CHs with a duration of up to 1.1 s (comparable to the current relaxation time) have been obtained. Equilibria and pressure profiles were obtained by including kinetic and motional Stark effect (MSE) data using the code TRANSP. Agreement between calculated and measured neutron fluxes is obtained only when the fast-ion diffusion coefficient is set to very high values which results in considerable fast-ion redistribution and a broadened pressure profile. The MSE-only equilibrium fits are in good agreement with such kinetic fits. While CHs collapse due to tearing modes, bursty MHD activity coherent with electron temperature relaxation events is observed when the CHs are sustained. Some collapse events also appear to be coincident with edge localized modes. The CH phase seems to have been limited by the no-wall beta limit. In the present set of discharges, wider CHs (steeper temperature gradients) appear to collapse at somewhat lower β. Tearing modes are seen to cause shrinking of the CH. While early electron heating is strongly dependent on the timing and power of the neutral beam and by the ECH power, it is seen that larger neutral beam powers are correlated with wider CHs which do not decay significantly. An extensive and perhaps difficult analysis of current transport is required to check whether a neoclassical poloidal flux evolution (diffusive EMF) is either consistent with a stationary current profile or that an additional or anomalous EMF is present.

Attainment of High Normalized Current by Current Profile Manipulation in the Pegasus Toroidal Experiment

Large stable values of normalized current IN are achievable in spherical tori due to the large gradient of the toroidal field across the plasma. This allows access to high Troyon beta limits. Values of IN > 12 MA/m-T are expected to be stable to ideal MHD modes in the ultra-low-A Pegasus Toroidal Experiment. In previous experiments, the toroidal field utilization (Ip/Itf) was found to be limited roughly to unity by the onset of large-scale low-order tearing modes during the current ramp, which limited IN to roughly 6 MA/m-T. Three techniques have now been developed that allow access to higher values of IN, primarily through modification of the current density profile. The first of these techniques employs a fast rampdown of the low-inductance toroidal field coil to rapidly decrease Itf. The other techniques employ gas-fueled washer stack guns as plasma sources. The second technique uses these guns to provide preionization at low toroidal field, while the third relies on the guns as helicity sources to form ST plasmas non-inductively. Using these techniques, values of Ip/Itf > 2 have been obtained.

Optimization of sensor signals for resistive wall mode control in ITER

Active control of the resistive wall mode has been considered as an alternative way, besides passive control by plasma rotation, to stabilize the mode in advanced scenarios in ITER. We show that a significant improvement of the feedback system is achievable by optimizing the detection system, without changing the active coil system in the design. Both analytical theory in cylindrical geometry, and toroidal calculations for ITER equilibria, show that the resistive wall mode can be stabilized for plasma pressures close to the ideal-wall beta limit, using the new technique with even radial sensors. This high plasma pressure is stabilized with the feedback coils outside the vacuum vessel, as in the nominal ITER design.

Chapter 9: ITER contributions for Demo plasma development

The chapter summarizes the physics issues of the demonstration toroidal fusion power plant (Demo) that can be addressed by ITER operation. These include burning plasma specific issues, i.e. energetic particle behaviour and plasma self-heating effects, and a broader class of power-plant scale physics issues that cannot be fully resolved in present experiments. A critical issue for Demo is whether MHD and energetic particle modes driven by fast particles will become unstable and affect plasma performance. Self-heating effects are expected to be especially important for control of steady-state plasmas with internal transport barriers (ITBs) and high bootstrap current fractions. Experimental data from ITER will improve strongly the physics basis of projections to Demo of major plasma parameters such as the energy confinement time, beta and density limits, edge pedestal temperature and density, and thermal loads on in-vessel components caused by ELMs and disruptions. ITER will also serve as a test bed for fusion technology studies, such as power plant plasma diagnostics, heating and current drive systems, plasma facing components, divertor and blanket modules. Finally, ITER is expected to provide benefits for the understanding of burning plasma behaviour in other magnetic confinement schemes.

Coupling between Radial and Torsional Oscillations in a Magnetized Plasma and Possible Stellar Applications

The weakly non-linear interaction of slow magnetoacoustic and torsional Alfven waves propagating along an unperturbed poloidal magnetic field in a stellar interior is studied in the high plasma beta limit. It is shown that slow magnetoacoustic waves parametrically drive torsional modes with the half frequency and wave number. Therefore global slow magnetoacoustic waves which have a radial velocity polarization, may amplify the torsional oscillations. Possible applications of this mechanism to the Sun and binary stars are briefly discussed.

Observation of Equilibria with a Double Magnetic Axis in LHD

Finite-beta equilibria with a double magnetic axis have been realized in the Large Helical Device (LHD). Since the rotational transform is weak in the central region of the LHD, the effect of an externally applied quadrupole field is more pronounced in the central region than in the periphery. Consequently, the magnetic axis splits due to a moderate elongation. In the case of vertical elongation, the figure-eight structure of the magnetic surfaces has been observed in a soft X-ray image. Degradation of confinement due to the appearance of the separatrix is suggested in the case of horizontal elongation, which is closely related to the equilibrium beta limit. The three-dimensional magnetohydrodynamic equilibrium code HINT, which does not assume the existence of nested flux surfaces, provides physical pictures consistent with the experimental observations.