High-beta Equilibria Research Articles

Experimental Study of High-beta Equilibrium Transition of ST Plasmas during Merging/Reconnection Heating

Experimental Study of High-beta Equilibrium Transition of ST Plasmas during Merging/Reconnection Heating

Kinetic construction of the high-beta anisotropic-pressure equilibrium in the magnetosphere

A theoretical model of the high-beta equilibrium of magnetospheric plasmas was constructed by consistently connecting the (anisotropic pressure) Grad–Shafranov equation and the Vlasov equation. The Grad–Shafranov equation was used to determine the axisymmetric magnetic field for a given magnetization current corresponding to a pressure tensor. Given a magnetic field, we determine the distribution function as a specific equilibrium solution of the Vlasov equation, using which we obtain the pressure tensor. We need to find an appropriate class of the distribution function for these two equations to be satisfied simultaneously. Here, we consider the distribution function that maximizes the entropy on the submanifold specified by the magnetic moment. This is equivalent to the reduction of the canonical Poisson bracket to the noncanonical one having the Casimir corresponding to the magnetic moment. The pressure tensor then becomes a function of the magnetic field (through the cyclotron frequency) and flux function, satisfying the requirement of the Grad–Shafranov equation.

Theoretical studies of equilibrium beta limit in LHD plasmas

This study deals with the equilibrium beta limit for high-beta plasmas of the Large Helical Device using a 3D equilibrium calculation code, HINT, which is an initial value solver based on the relaxation method without an assumption of nested flux surfaces. For the finite beta equilibrium, due to the nonlinear 3D equilibrium response, the flux surface breaking begins, and the magnetic field line becomes stochastic in the peripheral region. However, although the magnetic axis shifts until the conventional theoretical beta limit, the separatrix limiting the equilibrium does not appear in the plasma core. For the high beta equilibrium, breaking of the force balance begins in the strongly stochastized region. To keep the force balance in that region, the pressure gradient decreases and the fixed profile is relaxed. As a result, the volume averaged beta, which is equivalent to the plasma stored energy, saturates, although the peak beta increases. The beta value, where the force balance begins breaking in the inside of the stochastic region, is proposed as a new index of the equilibrium beta limit for the typical heliotron-type stellarator. According to this new index of the equilibrium beta limit, the achieved beta value in the experiment of the Large Helical Device is still lower than the equilibrium beta limit. This suggests that if the heating power can be increased, the achieved beta value may still increase.

High-beta analytic equilibria in circular, elliptical, and D-shaped large aspect ratio axisymmetric configurations with poloidal and toroidal flows

The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [E. Hameiri, Phys. Plasmas 20, 024504 (2013)], analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. To extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [L. Guazzotto et al., Phys. Plasmas 11(2), 604–614 (2004)] is presented for relevant scenarios.

High-beta equilibria in tokamaks with pressure anisotropy and toroidal flow

We extend previous analytical calculations of 2D high-β equilibria in order-unity aspect ratio tokamaks with toroidal flow to include pressure anisotropy, assuming guiding-center theory for a bi-Maxwellian plasma and the ideal MHD Ohm's law. Equilibrium solutions are obtained in the core region (which fills most of the plasma volume) and the boundary layer. We find that pressure anisotropy with p∥>p⊥ ( p∥<p⊥) reduces (enhances) the plasma diamagnetism relative to the isotropic case whenever an equilibrium solution exists. Sufficiently fast toroidal flows ( Ω>Ωmin) were previously found to suppress the field-free region (diamagnetic hole) that exists in static isotropic high-β equilibria. We find that all equilibrium solutions with pressure anisotropy suppress the diamagnetic hole. For the static case with a volume-averaged toroidal beta of 70%, plasmas with max(p∥/p⊥)>α1=1.07 have equilibrium solutions. We find that α1 decreases with increasing toroidal flow speed, and above the flow threshold Ωmin we find α1=1, so that all p∥>p⊥ plasmas have equilibrium solutions. On the other hand, for p∥<p⊥ there are no equilibrium solutions below Ωmin. Above Ωmin (where there is no diamagnetic hole in the isotropic case), equilibrium solutions exist for α2<min(p∥/p⊥)<1, where α2 decreases from unity with increasing flow speed. The boundary layer width increases and the Shafranov shift decreases for p∥>p⊥, while the converse is true for p∥<p⊥.

KSTAR equilibrium operating space and projected stabilization at high normalized beta

Along with an expanded evaluation of the equilibrium operating space of the Korea Superconducting Tokamak Advanced Research, KSTAR, experimental equilibria of the most recent plasma discharges were reconstructed using the EFIT code. In near-circular plasmas created in 2009, equilibria reached a stored energy of 54 kJ with a maximum plasma current of 0.34 MA. Highly shaped plasmas with near double-null configuration in 2010 achieved H-mode with clear edge localized mode (ELM) activity, and transiently reached a stored energy of up to 257 kJ, elongation of 1.96 and normalized beta of 1.3. The plasma current reached 0.7 MA. Projecting active and passive stabilization of global MHD instabilities for operation above the ideal no-wall beta limit using the designed control hardware was also considered. Kinetic modification of the ideal MHD n = 1 stability criterion was computed by the MISK code on KSTAR theoretical equilibria with a plasma current of 2 MA, internal inductance of 0.7 and normalized beta of 4.0 with simple density, temperature and rotation profiles. The steep edge pressure gradient of this equilibrium resulted in the need for significant plasma toroidal rotation to allow thermal particle kinetic resonances to stabilize the resistive wall mode (RWM). The impact of various materials and electrical connections of the passive stabilizing plates on RWM growth rates was analysed, and copper plates reduced the RWM passive growth rate by a factor of 15 compared with stainless steel plates at a normalized beta of 4.4. Computations of active RWM control using the VALEN code showed that the n = 1 mode can be stabilized at normalized beta near the ideal wall limit via control fields produced by the midplane in-vessel control coils (IVCCs) with as low as 0.83 kW control power using ideal control system assumptions. The ELM mitigation potential of the IVCC, examined by evaluating the vacuum island overlap created by resonant magnetic perturbations, was analysed using the TRIP3D code. Using a combination of all IVCCs with dominant n = 2 field and upper/lower coils in an even parity configuration, a Chirikov parameter near unity at normalized poloidal flux 0.83, an empirically determined condition for ELM mitigation in DIII-D, was generated in theoretical high-beta equilibria. Chirikov profile optimization was addressed in terms of coil parity and safety factor profile.

Low-Frequency Oscillations of Plasma in the Gas Dynamic Trap

Vortex confinement is a new and effective method of convective losses suppression in gas-dynamic trap. Low frequency oscillations of radial magnetic field were observed during study of plasma MHD activity. Experimental data confirm the main statements of the vortex confinement theory. We also observed displacement of fast ions turning point. It can be explained by high beta equilibrium effects.

Helical Pitch Parameter Dependency of High Beta Equilibrium of Helical Plasmas

The helical pitch parameter (γ) dependency of equilibrium and the stability of the high beta plasma in the LHD type magnetic configuration is studied numerically. It is confirmed that the small γ configurations are favorable for the LHD-type fusion reactors in the point of robustness of high beta equilibrium, compatibility of easy ignition and high output power of core plasma, in addition to a sufficient space for blankets.

Sustainment of FRC-equilibrium by use of a centre solenoid in TS-4

This paper addresses a basic equilibrium/confinement question whether a field reversed configuration (FRC) with a balanced amount of magnetic and thermal energies is sustained solely by inductive magnetic flux (magnetic energy) injection of a centre solenoid coil. A new finding is that preferential injection of magnetic energy causes a significant increase in thermal energy of the FRC to maintain high-beta equilibrium. The FRC plasma is concluded to have the robustness needed to self-adjust the plasma heating power depending on the magnetic energy injection. In the flux ramped-up phase, i.e. in the phase where the applied inductive electric field did not penetrate into the core of the plasma, the J × B compression (pinch effect) was considered to be dominant for heating the FRC. On the other hand, after the applied inductive electric field penetrated into the core region the resistivity of the FRC increased up to 10–20 times larger than the classical spitzer resistivity. This anomalous heating was the most probable cause for sustainment of the high-beta FRC in this phase. A quasi-steady state of the high-beta (volume averaged beta ∼0.6–0.7) FRC for 0.15 ms (∼energy confinement time of conventional FRCs) was achieved by the use of the present low power centre solenoid current drive. This successful result leads us to the next stage steady sustainment experiment of an FRC by increased capacitor bank energy.

RT-1 Project: Magnetosphere-Like Plasma Experiment

The Ring Trap-1 (RT-1) is a novel plasma device designed to explore various effects of plasma flow in the most universal and fundamental magnetosphere-like configuration. A super-conducting ring magnet, levitated in the vacuum chamber, produces a dipole magnetic field that traps high-temperature plasma. Plasma is produced by electron cyclotron heating using an 8.2GHz microwave. The mechanism of plasma confinement is based on the theory of high-beta equilibrium that is self-organized in a flowing plasma. Realization of such a configuration in a laboratory system, which is known to exist in some astronomical systems, may open a way to the advanced-fuel fusion. Analyses of the equilibrium and stability pose interesting theoretical and experimental challenges.

First Plasma in the RT-1 Device

The first plasma experiment using the Ring Trap-1 (RT-1) was performed on January 12, 2006. The RT-1 is a novel plasma device constructed to explore ways to the advanced-fuel fusion. The mechanism of plasma confinement is based on the theory of high-beta equilibrium that is self-organized in a flowing plasma. A superconducting ring, levitated in the vacuum chamber, produces a magnetic field that traps high-temperature plasma, creating a magnetosphere-like configuration. Plasma is produced by electron cyclotron heating using an 8.2 GHz microwave.

Theoretical considerations of doublet-like configuration in LHD

In order to investigate the separatrix structure inside the plasma, the magnetohydrodynamic (MHD) equilibrium and the high-energy particle orbit are studied for the Large Helical Device plasmas with two magnetic axes located within the two lobes of a figure-eight magnetic separatrix. Magnetic axes are split for strongly elongated cross section by the external quadrupole field. MHD equilibria without assuming existence of nested flux surfaces are obtained from the HINT code. For the vertical elongation, the figure-eight structure evolves due to the finite beta effect and the structure of magnetic field lines outside the separatrix is ergodized. For the horizontal elongation, the Shafranov shift is very large Δ/a > 0.5. To see MHD equilibrium beta limit, high beta equilibrium (∼5%) is studied. The magnetic axis shifts extremely to the outside of the torus and a new figure-eight separatrix appears. The particle orbit on the separatrix is studied by solving the guiding-centre drift equation on the rectangular grid. If the reflecting point of the trapped particle is close to the separatrix, the banana width is very large implying large, nonlocal neoclassical transport.

Analytic description of high poloidal beta equilibrium with a natural inboard poloidal field null

Analytical high poloidal beta equilibria for toroidally axisymmetric plasmas with arbitrary aspect ratio and elongation are described. These equilibria that can describe a transition from nondivertor to divertor configuration are exact solutions of the Grad-Shafranov equation when the toroidal current density is quasiuniform. Generally, these are high poloidal beta equilibria, limited by the appearance of a natural inboard poloidal field null. Some of their properties, including the nonuniformity of the poloidal magnetic field in the poloidal direction, the safety factor profile and the magnetic shear profile near the separatrix, the parameter dependence of the poloidal beta βp and εβp, as well as the toroidal beta βT on the aspect ratio and the elongation of the magnetic surface, are discussed. Applications to experiments of the Tokamak Fusion Test Reactor (TFTR) [Sabbagh et al., Phys. Fluids B3, 2277 (1991)] are particularly analyzed.

Contour dynamics method for solving the Grad–Shafranov equation with applications to high beta equilibria

Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of a plasma confined by an axisymmetric magnetic field. Nevertheless, they are limited to low beta or small plasma pressure. Combining a nonconservative variational principle with a contour dynamics approach, the approach presented in this paper converges for extreme high beta configurations. By reducing the dimension of the problem from two to one, a compact and efficient numerical algorithm can be developed, and a wide range of boundary shapes can be utilized. Furthermore, the iterative nature of this technique greatly facilitates convergence at high beta while minimizing computation times.

Confinement characteristics of the quasi-axisymmetric stellarator CHS-qa

A low aspect ratio stellarator CHS-qa with a quasi-axisymmetric configuration has been designed at NIFS, whose main objective is the experimental study of confinement improvement. The high beta equilibrium was obtained with the HINT code which does not require the assumption of nested magnetic surfaces. The equilibria with bootstrap current were calculated using the SPBSC code and their global MHD stability was evaluated with the TERPSICHORE code. The boundary structure of the magnetic field was examined to scope divertor options. A new method of calculating the neoclassical transport properties was developed for toroidal configurations with very small non-axisymmetry. The control knobs of the configuration properties are proposed for the study of improved confinement.

Beltrami fields in plasmas: High-confinement mode boundary layers and high beta equilibria

A general solenoidal vector field, such as a magnetic field or an incompressible flow, can be decomposed into an orthogonal sum of Beltrami fields (eigenfunctions of the curl operator). Nonlinear dynamics of a plasma induces complex couplings among these Beltrami fields. In a single-fluid magnetohydrodynamic (MHD) plasma, however, the energy condensates into a single Beltrami magnetic field resulting in the self-organization of a force-free equilibrium, that is, the Taylor relaxed state. By relating the velocity and the magnetic fields, the Hall term in the two-fluid model leads to a singular perturbation that enables the formation of an equilibrium given by a pair of two different Beltrami fields. This new set of relaxed states, despite the simple mathematical structure, includes a variety of plasma states that could explain a host of interesting phenomena. The H-mode (high-confinement) boundary layer, where a diamagnetic structure is self-organized under the coupling of the magnetic field, flow, electric field, and pressure, is an example. The theory also predicts the possibility of producing high beta equilibrium.

Conceptual design of a breeding blanket with super-heated steam cycle for CREST-1

Conceptual design of a tritium breeding blanket for CREST-1 was conducted. CREST-1 (Compact REversed Shear Tokamak), a conceptual design of water-cooled commercial reactor based on a reversed shear high beta equilibrium, has been proposed as a possible scenario to an economical and feasible reactor succeeding the ITER project. In the present design study, a possibility of cost competitive fusion power plants with water-cooled concept, which has much experience in nuclear power plants, was examined. The new blanket design is based on the low activation ferritic steel components and an advanced super-heated steam cycle, which is used to realize a high thermal efficiency. High value of the thermal efficiency is very effective for reduction of the cost of electricity. On designing the blanket, allowable temperature range of the structure material, low activation ferritic steel is assumed to be 350–900 K with an expectation of the material research and development. Mixture of lithium oxide pebbles and beryllium pebbles is installed in the breeding zone for high tritium breeding ratio and high thermal conductivity of the breeding zone. Mixture ratio of beryllium and lithium-6 enrichment were optimized from viewpoints of temperature distribution in the breeding zone, achievable tritium breeding ratio and its reduction due to burn up. The designed blanket system has approximately 1.4 of local tritium breeding ratio with a 1.0 mm thickness of zirconium plate which is placed at 24 cm from the plasma surface as a conducting shell for kink stabilization. Arrangements of cooling channels and breeding zones and flow rate and inlet temperature of the coolant were also optimized to keep the temperatures of structure materials, breeding materials and coolant in the allowable range. The first wall is cooled by pressurized water at about 570 K. The coolant out of cooling channels of the first wall is lead to those of breeding zone and starts partially boiling. The steam is super-heated up to 750 K in the blanket. This high temperature raises the thermal efficiency of turbine to 41%. Our cost assessment has shown that CREST-1 generates about 1.16 GWe electric power within a competitive COE range.

Stability of short wavelength tearing and twisting modes

The stability and mutual interaction of tearing and twisting modes in a torus is governed by matrices that generalize the well-known Δ′ stability index. The diagonal elements of these matrices determine the intrinsic stability of modes that reconnect the magnetic field at a single resonant surface. The off-diagonal elements indicate the strength of the coupling between the different modes. A method is presented for evaluating the elements of these matrices, in the limit of short wavelength, from the free energy driving radially extended ballooning modes. This method is applied to calculate the tearing and twisting Δ′ for a model high-beta equilibrium with circular flux surfaces.

Study of a compact reversed shear Tokamak reactor

A reversed shear configuration, which was observed recently in some Tokamak experiments, might have a possibility to realize compact and cost-competitive Tokamak reactors. In this study, a compact (low cost) commercial reactor based on the shear reversed high beta equilibrium with β N=5.5, is considered, namely the Compact Reversed Shear Tokamak, CREST-1. The CREST-1 is designed with a moderate aspect ratio ( R/a=3.4), which will allow us to experimentally develop this CREST concept by ITER. This will be very advantageous with regard to the fusion development strategy. The current profile for the reversed shear operation is sustained and controlled in steady state by bootstrap (88%), beam and rf driven currents, which are calculated by a neo-classical model code in 3D geometry. The MHD stability has been checked by an ideal MHD stability analysis code (ERATO) and it has been confirmed that the ideal low n kink, ballooning and Mercier modes are stable while a closed conductive shell is required for stability. Such a compact Tokamak can be cost-competitive as an electric power source in the 21st century and it is one possible scenario in realizing a commercial fusion reactor beyond the ITER project.

Free-boundary, high-beta equilibrium in a large aspect ratio tokamak with nearly circular plasma boundary

An analytic solution is obtained for free-boundary, high-beta equilibria in large aspect ratio tokamaks with a nearly circular plasma boundary. In the absence of surface currents at the plasma-vacuum interface, the free-boundary equilibrium solution introduces constraints arising from the need to couple to an external vacuum field which is physically realizable with a reasonable set of external field coils. This places a strong constraint on the pressure profiles that are consistent with a given boundary shape at high εβp. The equilibrium solution also provides information on the flux surface topology. The plasma is bounded by a separatrix. Increasing the plasma pressure at fixed total current causes the plasma aperture to decrease in a manner that is described.