Poloidal Beta Value Research Articles

Development of an Advanced-Fuel Nuclear Fusion Experiment

As a first step to exploring the possibilities of D-3He plasmas, a deuterium-tritium burning plasma experiment at high field and plasma densities, which can be much closer to the required parameters than present-day experiments, is particularly attractive. Compact high-field experiments were the first to be proposed in order to achieve fusion ignition conditions based on existing technology and the known properties of high-density plasmas. In previous studies, a feasibility study of a high-field D-3He experiment of larger dimensions and higher fusion power than Ignitor, but based on Ignitor technologies, was brought to the proposed Candor fusion experiment. Unlike Ignitor, Candor would operate with values of poloidal beta around unity and the central part of the plasma column in the second stability region. The toroidal field coils are divided into two sets of coils, and the central solenoid (air core transformer) is placed between them in the inboard part. In this paper, a revised design of Candor is proposed, based on the new technologies. This tokamak is capable of reaching D-3He ignition on the basis of existing technologies and knowledge of plasma, without any optimistic extrapolation.

Plasma equilibrium based on EC-driven current profile with toroidal rotation on QUEST

In the EC-driven (8.2 GHz) steady-state plasma on QUEST, plasma current seems to flow in the open magnetic surface in the outside of the closed magnetic surface in the low-field region according to plasma current fitting method without taking equilibrium into account. In our previous work, plasma equilibrium solution was fitted assuming all plasma current is flowing in the inside of the Last Closed Flux Surface (LCFS). It was solved within isotropic pressure profile by EFIT code. Opposite-polarity current density region appeared in the high-field region. Here in this article, considering the toroidal rotation, the equilibrium is fitted within nested magnetic surfaces by SU-EFIT. Though the plasma magnetic axis shifts outward due to the centrifugal force, the opposite polarity current does not disappear in the high-field region. And relation between the toroidal rotation speed and the poloidal beta value will be discussed.

Influence of plasma pedestal profiles on access to ELM-free regimes in ITER

The influence of current density and pressure gradient profiles in the pedestal on the access to the regimes free from edge localized modes (ELMs) like quiescent H-mode in ITER is investigated. Using the simulator of MHD modes localized near plasma boundary based on the KINX code, calculations of the ELM stability were performed for the ITER plasma in scenarios 2 and 4 under variations of density and temperature profiles with the self-consistent bootstrap current in the pedestal. Low pressure gradient values at the separatrix, the same position of the density and temperature pedestals and high poloidal beta values facilitate reaching high current density in the pedestal and a potential transition into the regime with saturated large scale kink modes. New version of the localized MHD mode simulator allows one to compute the growth rates of ideal peeling-ballooning modes with different toroidal mode numbers and to determine the stability region taking into account diamagnetic stabilization. The edge stability diagrams computations and sensitivity studies of the stability limits to the value of diamagnetic frequency show that diamagnetic stabilization of the modes with high toroidal mode numbers can help to access the quiescent H-mode even with high plasma density but only with low pressure gradient values at the separatrix. The limiting pressure at the top of the pedestal increases for higher plasma density. With flat density profile the access to the quiescent H-mode is closed even with diamagnetic stabilization taken into account, while toroidal mode numbers of the most unstable peeling-ballooning mode decrease from n = 10−40 to n = 3−20.

Tokamak elongation – how much is too much? Part 2. Numerical results

The analytic theory presented in Paper I is converted into a form convenient for numerical analysis. A fast and accurate code has been written using this numerical formulation. The results are presented by first defining a reference set of physical parameters based on experimental data from high performance discharges. Scaling relations of maximum achievable elongation (${\it\kappa}_{max}$) versus inverse aspect ratio (${\it\varepsilon}$) are obtained numerically for various values of poloidal beta (${\it\beta}_{p}$), wall radius $(b/a)$ and feedback capability parameter (${\it\gamma}{\it\tau}_{w}$) in ranges near the reference values. It is also shown that each value of ${\it\kappa}_{max}$ occurs at a corresponding value of optimized triangularity (${\it\delta}$), whose scaling is also determined as a function of ${\it\varepsilon}$. The results show that the theoretical predictions of ${\it\kappa}_{max}$ are slightly higher than experimental observations for high performance discharges, as measured by high average pressure. The theoretical ${\it\delta}$ values are noticeably lower. We suggest that the explanation is associated with the observation that high performance involves not only $n=0$ MHD stability, but also $n\geqslant 1$ MHD modes described by ${\it\beta}_{N}$ in the Troyon limit and transport as characterized by ${\it\tau}_{E}$. Operation away from the $n=0$ MHD optimum may still lead to higher performance if there are more than compensatory gains in ${\it\beta}_{N}$ and ${\it\tau}_{E}$. Unfortunately, while the empirical scaling of ${\it\beta}_{N}$ and ${\it\tau}_{E}$ with the elongation (${\it\kappa}$) has been determined, the dependence on ${\it\delta}$ has still not been quantified. This information is needed in order to perform more accurate overall optimizations in future experimental designs.

Semi-analytic approach for determination of poloidal beta limits using plasma internal inductance in Damavand tokamak

The Grad–Shafranov equation plays a very important role in analysis of the plasma equilibrium in magnetic confinement configurations such as the tokamak. In tokamaks which are operating in Ohmic heating regime, the Grad–Shafranov equation can be expanded through the inverse aspect ratio parameter. Consequently, the first order of poloidal flux function and the poloidal/radial components of the magnetic field are obtained. In this study, by considering the effect of tokamak non-circularity, a semi-analytic model which is based on magnetic measurements was developed and highest and lowest values of plasma internal inductance and poloidal beta were estimated by means of the poloidal and radial magnetic field experimental data acquired from Damavand tokamak magnetic probes.

Improved confinement region without large magnetohydrodynamic activity in TPE-RX reversed-field pinch plasma

We found that spontaneous improved confinement was brought about depending on the operating region in the Toroidal Pinch Experiment-Reversed eXperiment (TPE-RX) reversed-field pinch plasma [Y. Yagi et al., Fusion Eng. Des. 45, 421 (1999)]. Gradual decay of the toroidal magnetic field at plasma surface Btw reversal makes it possible to realize a prolonged discharge, and the poloidal beta value and energy confinement time increase in the latter half of the discharge, where reversal and pinch parameters become shallow and low, respectively. In the latter half of the discharge, the plasma current and volume-averaged toroidal magnetic field 〈Bt〉 increase again, the electron density slowly decays, the electron temperature and soft X-ray radiation intensity increase, and the magnetic fluctuations are markedly reduced. In this period of improved confinement, the value of (〈Bt〉-Btw)/Bpw, where Bpw is the poloidal magnetic field at the plasma surface, stays almost constant, which indicates that the dynamo action occurs without large magnetohydrodynamic activities.

Diamagnetic measurements in the STOR-M tokamak by a flux loop system exterior to the vacuum vessel

Diamagnetic measurements of poloidal beta have been performed in the STOR-M tokamak by a flux loop placed exterior to the vacuum chamber with compensation for the vacuum toroidal field using a nonenclosing coplanar coil, and vibrational compensation from auxiliary coils. It was found that in STOR-M conditions (20% toroidal magnetic field decay over discharge) there is significant influence on the diamagnetic flux measurements from strong residual signals, presumably from image currents being induced by the toroidal field coils, requiring further compensation. A blank (nonplasma) shot is used specifically to eliminate the residual component which is not proportional to the toroidal magnetic field. Data from normal Ohmic discharge operation is presented and calculations of poloidal beta from coil data (beta(theta) approximately 0.5) is found to be in reasonable agreement with the values of poloidal beta obtained from measurements of electron density and Spitzer temperature with neoclassical corrections for trapped electrons. Contributions present in the blank shot (residual) signal and the limitations of this method are discussed.

Scaling of bootstrap current on equilibrium and plasma profile parameters in tokamak plasmas

A detailed study of the bootstrap current dependence upon some plasma profile parameters is performed and the relations between this intrinsic current with plasma equilibrium quantities such as internal plasma inductance, normalized and poloidal beta values, loop voltage and central safety factor are analysed within an equilibrium calculation valid for arbitrary aspect ratio tokamaks. In the equilibrium model, the total plasma current is given by the sum of the diamagnetic, Pfirsch–Schlüter and the neoclassical ohmic and bootstrap currents. We considered here only cases where the bootstrap current is generated by thermal particles and its profile is calculated by using the fitted formulae proposed by Sauter et al. Strong anisotropies and deep gradients (related to the presence of internal transport barriers) in the plasma pressure profiles are not taken into account. We have also accounted in our analysis for variations of Zeff, plasma elongation, triangularity, magnetic field and plasma current. This study is performed for the spherical tokamak Experimento Tokamak Esférico (ETE), and estimates obtained for the bootstrap current fraction (Ibs/Ip) are compared with a general empirical scaling law proposed by Hoang et al. Due to some deviations observed between Hoang's scaling and our results, a new expression for Ibs/Ip, given in terms of the normalized beta, cylindrical q, internal inductance parameter (li) and peakedness of pressure profile (cp), is proposed. Moreover, a second expression given as a combination of these parameters described in terms of βpol and cp is also presented. Both formulations were obtained from a least-square fitting upon about 350 ETE equilibria, in which some stability criteria were taken into account. Errors mostly up to the order of 10% are obtained when both expressions are compared with the equilibrium estimates for the bootstrap current in ETE.

Physics of strong internal transport barriers in JT-60U reversed-magnetic-shear plasmas

The physics of strong internal transport barriers (ITBs) in JT-60U reversed-magnetic-shear (RS) plasmas has been studied through the modelling on the 1.5 dimensional transport simulation. The key physics to produce two scalings on the basis of the JT-60U box-type ITB database are identified. As for the scaling for the narrow ITB width proportional to the ion poloidal gyroradius, the following three physics are important: (1) the sharp reduction of the anomalous transport below the neoclassical level in the RS region, (2) the autonomous formation of pressure and current profiles through the neoclassical transport and the bootstrap current and (3) the large difference between the neoclassical transport and the anomalous transport in the normal-shear region. As for the scaling for the energy confinement inside ITB (ϵfβp,core ≈ 0.25, where ϵf is the inverse aspect ratio at the ITB foot and βp,core is the core poloidal beta value), the value of 0.25 is found to be a saturation value due to the MHD equilibrium. The value of ϵfβp,core reaches the saturation value, when the box-type ITB is formed in the strong RS plasma with a large asymmetry of the poloidal magnetic field, regardless of the details of the transport and the non-inductively driven current.

Soft x-ray tangential imaging of the NSTX core plasma by means of an MPGD pinhole camera

A fast x-ray system based on a micro pattern gas detector has been used for the first time, to investigate emission from the plasma core of the National Spherical Tokamak eXperiment (NSTX), at the Princeton Plasma Physics Laboratory. The results presented in this work demonstrate the capability of such a device to measure with a time resolution of the order of 1 ms, the curvature and the elongation of the x-ray iso-emissivity contours, under various plasma conditions. Also, comparisons with the magnetic surface structure calculated by the EFIT code show good agreement between reconstructed flux surface and the soft x-ray (SXR) emissions for poloidal beta values up to 0.6. For greater values of beta, x-ray iso-emissivity contours become circular, while magnetic flux surface reconstructions yield elongation 1.5 < k < 2.2.The x-ray images have been acquired with a (statistical) signal to noise ratio (SNR) per pixel of about 30. Thanks to the direct and efficient x-ray conversion and its operation in a photon counting mode, this new diagnostic tool allows the routine investigation of the plasma core with a sampling rate of 1 kHz and extremely high SNR under all experimental conditions in NSTX.

Aspect ratio dependence of the ideal internal kink mode stability in a toroidal plasma with circular cross section

The theory by Bussac et al. [Phys. Rev. Lett. 35, 1638 (1975)] on the stability of the ideal, internal kink mode in a toroidal plasma with circular cross section is extended to finite values of the inverse aspect ratio εa. The aspect ratio dependence comes from the term following the Bussac term in a large aspect ratio expansion of the potential energy δW of the mode. Although formally of higher order in the inverse aspect ratio, this term is of the same order of magnitude as the Bussac term when the safety factor q is close to (but below) unity at the plasma center and εa is finite. The new term is always stabilizing, but affects δW only marginally when the poloidal beta value βp is small and/or εa is small. For values of εa relevant for most tokamaks and βp larger than 0.3–0.5, however, the new term strongly modifies δW and leads to a second stability regime of the ideal, internal kink mode, in agreement with earlier numerical calculations.

Comparison of confinement in resistive-shell reversed-field pinch devices with two different magnetic shell penetration times

A loop voltage is required to sustain the reversed-field pinch (RFP) equilibrium. The configuration is characterized by redistribution of magnetic helicity but with the condition that the total helicity is maintained constant. The magnetic field shell penetration time, τs, has a critical role in the stability and performance of the RFP. Confinement in the EXTRAP device has been studied with two values of τs, first (EXTRAP-T2) with τs of the order of the typical relaxation cycle timescale and then (EXTRAP-T2R) with τs much longer than the relaxation cycle timescale, but still much shorter than the pulse length. Plasma parameters show significant improvements in confinement in EXTRAP-T2R. The typical loop voltage required to sustain comparable electron poloidal beta values is a factor of 3 lower in the EXTRAP-T2R device. The improvement is attributed to reduced magnetic turbulence.

Turbulence trigger for neoclassical tearing modes in tokamaks

A stochastic trigger by micro-turbulence for neoclassical tearing modes (NTMs) is studied. NTMs induce topological change of magnetic structure and have a subcritical nature. It is found that the NTM can be excited regardless of the presence of seed islands. The transition rate of the NTM, its probability density function and its statistically averaged amplitude are obtained. The boundary in the phase diagram is determined as the statistical long time average of the transition conditions. The NTM can be excited by crossing this boundary even in the absence of other global instabilities. The boundary is expressed in terms of the poloidal beta value. Its dependence on the ratio between the ion banana width and radius of the rational surface, ρb/rs, is found to vary in the range ρb/rs–(ρb/rs)2, depending on the linear stability of the tearing mode.

Upgrade of the present JET shape and vertical stability controller

The development of linear model of the present JET (Joint European Torus) electromagnetic system has allowed the design of an optimized Plasma Shape Controller. Quite extreme plasma boundaries (elongation k=1.9 and triangularity 〈 δ〉=0.6) are shown to be controllable also in the presence of large variations of the values of poloidal beta (0.1< β p<0.6) and/or internal magnetic inductance (0.8< l i<1.4). The developed model gives also an excellent simulation of the growth rate of the experimental vertical disruptions.

On the poloidal beta evolution of fixed-q heating in tokamaks

For typical tokamak safety factor, q, profiles and aspect ratio, A, fixed-q heating leads to a limit for equilibrium code convergence near a poloidal beta value of 1+A. Also presented is a simple tanh functional fit that saturates just above the 1+A limit. This poloidal beta limit does not depend on the q-profile, but the total beta achieved at this critical poloidal beta does depend strongly on q. In this paper we confirm very early computations of the flux-conserving tokamak that routinely achieved a poloidal beta &amp;gt;A.

Recent Progress in High-Beta Studies in the REP.

The recent progress in high-beta reversed-fieldpinch (RFP) studies is reviewed. Experimental efforts to suppress the RPF dynamo modes (core resonant tearing modes) have resulted in remarkable confinement improvement in large RFP experiments. The poloidal beta (≈total beta) value of ˜20% has been achieved with improved confinement. After suppressing the current-driven modes, the role of pressure-driven modes will become important in the RFP dynamics. Theoretical studies of pressure-driven modes have revealed that ideal-mode-stable RFP configurations with βP≈30% exist, and that unstable mode spectra are sensitive to local current profiles as well as to pressure profiles. The role of the Suydam criterion on the stability of ideal modes is discussed briefly. From an experimental point of view, it is essential to investigate the beta limit in the RFP by increasing heating power density under conditions of improved confinement.

Stabilization of the Mercier modes in a tokamak by toroidal plasma rotation

The stability of localized modes (Mercier modes) in a tokamak with a toroidally rotating plasma is analyzed within the framework of compressible, ideal magnetohydrodynamics. For equilibria with large aspect ratio, poloidal beta value of order unity, and isothermal magnetic surfaces, it is found that sonic, toroidal rotation provides a strongly stabilizing effect for the Mercier modes, similar to the stabilization recently found for the internal kink mode in a rotating plasma [Wahlberg and Bondeson, Phys. Plasmas 7, 923 (2000)]. A finite oscillation frequency (Brunt–Väisälä frequency), of the order of the sound frequency, is shown to be associated with each magnetic surface. If Γ&amp;gt;1, where Γ is the exponent in the equation of state, the rotation transforms the Mercier instabilities to stable oscillations at the local Brunt–Väisälä frequency associated with the magnetic surface where the mode is located. If the plasma satisfies an isothermal equation of state (Γ=1), however, the stability of the Mercier modes becomes sensitive to the profile of the toroidal flow. In this case, the rotation is found to be stabilizing if the kinetic energy density of the rotation is an increasing function of the minor radius. In the opposite case, the rotation is destabilizing unless the pressure profile is much more peaked than the kinetic energy density profile.

Analytic spherical torus plasma equilibrium model

An analytic spherical torus plasma equilibrium model is developed from a general solution to the Grad–Shafranov equation. The analytic model allows the calculation of axisymmetric plasma equilibria with arbitrary aspect ratio, elongation, triangularity, and diamagnetism. Using a numerical method, examples of optimized analytic equilibria are presented with plasma profiles similar to those of bootstrapped spherical tokamaks. Numerical results obtained from analytic equilibrium solutions for plasmas with aspect ratios A=1.4, elongations κ=3.0, and Troyon factors g=6.7 suggest that spherical tori can achieve toroidal beta values as large as βt≈35% at poloidal beta values βp≈1.53, with stable magnetic wells, W⩾0.

Isothermal internal kink instability in a toroidally rotating tokamak plasma

The properties of the internal kink instability in a tokamak with a toroidally rotating plasma is analysed from the ideal magnetohydrodynamic (MHD) equations, for plasmas satisfying an isothermal equation of state. The plasma equilibrium is assumed to have large aspect ratio, circular cross section, poloidal beta value ( β p) of order unity, and a rotational frequency comparable to the sound frequency. For an isothermal plasma, the rotational effect on the internal kink mode can be expressed in terms of a `rotational beta value' ( β Ω ) of the plasma. This quantity is defined in the same way as β p, with the kinetic energy density of the rotation ( u Ω ) replacing the plasma pressure. The rotation is found to be strongly stabilizing if u Ω is an increasing function of the minor radius ( β Ω<0 ). In the opposite case, the rotation is destabilizing unless the pressure profile is much more peaked than the kinetic energy density profile.

Analytical stability condition for the ideal m=n=1 kink mode in a toroidal plasma with elliptic cross section

An analytical stability condition for the ideal kink mode with toroidal mode number n=1 and (dominating) poloidal mode number m=1 in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity (e) is treated as a small parameter independent of the inverse aspect ratio (ε), and the expansion is performed up to order ε2e in the potential energy δW. It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when Δq=1−q0 is small (q0 is the safety factor at the magnetic axis), whereas an ellipticity of opposite sign (horizontal elongation) is stabilizing. By means of an additional expansion, in Δq, this effect is shown to be due to a Δq-independent term in δW proportional to the ellipticity and to the poloidal beta value (βp) at the q=1 surface. Since the leading order term in δW in the absence of ellipticity is of order Δq [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)], the ellipticity term dominates δW for sufficiently small values of Δq. The analytical result is in agreement with previous, numerical results [Lütjens et al., Nucl. Fusion 32, 1625 (1992)], and leads, already for rather moderately vertically elongated plasmas, to stability limits in βp much lower than the value 0.3 valid in the limit Δq→0 for a plasma with circular cross section.