Toroidal Current Density Profile Research Articles

Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad–Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

Self-organized magnetic equilibria in tokamak plasmas with very low edge safety factor

Tokamak plasmas often exhibit self-organizing behavior in which internal modes shape the toroidal current density profile, a common example being the sawtooth instability. However, such behavior has not been studied in detail for edge safety factor below 2 due to disruptive kink instabilities that typically prevent operation in this regime. Now, steady tokamak plasmas with an edge safety factor down to 0.8 have been created in the Madison Symmetric Torus, where disruptions are prevented due to a thick, conductive wall and a feedback power supply that sustains the plasma current. Internal measurements and nonlinear magnetohydrodynamic modeling reveal a family of safety factor profiles with a central value clamped near unity as the edge safety factor decreases, indicating current profile broadening through a relaxation process. As the safety factor decreases, the magnetic fluctuations become irregular, and the electron energy confinement time decreases.

Exploring the limits of Solov'ev profiles

Solov'ev profiles, which are the simplest that allow analytic solutions for the Grad–Shafranov (G–Sh) equation, can reproduce some features of experimental equilibria when the free parameters are appropriately chosen. The purpose of this paper is to study how the freedom in the choice of the parameters and the number of terms in a series solution to the homogeneous equation affect the equilibria obtained. Keeping equatorial symmetry for simplicity, the limits of this approach are tested. The behavior of the solutions for devices with different sizes and aspect ratios, such as ITER, ST40, and MAST-U, is studied, and also examples in which physically relevant figures of merit are found. While the toroidal current density profile is necessarily limited, and cannot be realistic, some of the main features of the magnetic field flux surfaces can be obtained. Three sets of parameters are distinguished, so their roles can be better understood: (1) The geometrical parameters of the boundary conditions, which determine the last closed magnetic field flux surface, (2) The general parameters of an experiment, such as the plasma current, the major radius, and the toroidal magnetic field at the major radius, and (3) the Solov'ev parameters. The regions of acceptable values for the Solov'ev parameters, are identified, depending on the topology of the solutions.

TRANSP-based closed-loop simulations of current profile optimal regulation in NSTX-Upgrade

Active control of the toroidal current density profile is critical for the upgraded National Spherical Torus eXperiment device (NSTX-U) to maintain operation at the desired high-performance, MHD-stable, plasma regime. Initial efforts towards current density profile control have led to the development of a control-oriented, physics-based, plasma-response model, which combines the magnetic diffusion equation with empirical correlations for the kinetic profiles and the non-inductive current sources. The developed control-oriented model has been successfully tailored to the NSTX-U geometry and actuators. Moreover, a series of efforts have been made towards the design of model-based controllers, including a linear-quadratic-integral optimal control strategy that can regulate the current density profile around a prescribed target profile while rejecting disturbances. In this work, the tracking performance of the proposed current-profile optimal controller is tested in numerical simulations based on the physics-oriented code TRANSP. These high-fidelity closed-loop simulations, which are a critical step before experimental implementation and testing, are enabled by a flexible framework recently developed to perform feedback control design and simulation in TRANSP.

Robust control of the current profile and plasma energy in EAST

Integrated control of the toroidal current density profile, or alternatively the q-profile, and plasma stored energy is essential to achieve advanced plasma scenarios characterized by high plasma confinement, magnetohydrodynamics stability, and noninductively driven plasma current. The q-profile evolution is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The MDE prediction depends heavily on the chosen models for the electron temperature, plasma resistivity, and non-inductive current drives. To aid control synthesis, control-oriented models for these plasma quantities are necessary to make the problem tractable. However, a relatively large deviation between the predictions by these control-oriented models and experimental data is not uncommon. For this reason, the electron temperature, plasma resistivity, and non-inductive current drives are modeled for control synthesis in this work as the product of an “uncertain” reference profile and a nonlinear function of the different auxiliary heating and current-drive (H&CD) source powers and the total plasma current. The uncertainties are quantified in such a way that the family of models arising from the modeling process is able to capture the q-profile and plasma stored energy dynamics from a typical EAST shot. A control-oriented nonlinear PDE model is developed by combining the MDE with the “uncertain” models for the electron temperature, plasma resistivity, and non-inductive current drives. This model is then rewritten into a control framework to design a controller that is robust against the modeled uncertainties. The resulting controller utilizes EAST's H&CD powers and total plasma current to regulate the q profile and plasma stored energy even when mismatches between modeled and actual dynamics are present. The effectiveness of the controller is demonstrated through nonlinear simulations.

Detection of an electron beam in a high density plasma via an electrostatic probe

An electron beam is detected by a 1D floating potential probe array in a relatively high density (1012–1013 cm−3) and low temperature (∼5 eV) plasma of the Magnetic Reconnection Experiment. Clear perturbations in the floating potential profile by the electron beam are observed. Based on the floating potential profile and a current balance equation to the probe array tips, the effective width of the electron beam is determined, from which we determine the radial and toroidal beam current density profiles. After the profile of the electron beam is specified from the measured beam current, we demonstrate the consistency of the current balance equation and the location of the perturbation is also in agreement with field line mapping. No significant broadening of the electron beam is observed after the beam propagates for tens of centimeters through the high density plasma. These results prove that the field line mapping is, in principle, possible in high density plasmas.

Analytic axisymmetric equilibria with pressure anisotropy and non-parallel flow

We extend previous work by two of the authors [Evangelias and Throumoulopoulos 2016 Plasma Phys. Control. Fusion 58 045022] and find analytic solutions to a generalized Grad–Shafranov equation (GGSE) with pressure anisotropy, hollow toroidal current density and incompressible flow of arbitrary direction. By specifying the surface-function terms involved in the GGSE as quadratic functions of the poloidal magnetic flux function, we derive analytic tokamak pertinent equilibria. It turns out that, irrespective of the shape of the toroidal current density profile, which is noticeably affected by the pressure anisotropy, this anisotropy has a paramagnetic effect for and a diamagnetic one for . In addition, the parallel flow induces paramagnetism, while the residual toroidal flow associated with the electric field induces diamagnetism. The analytic results obtained here contribute to the physics understanding of axisymmetric equilibria with pressure anisotropy and flow, and can be employed for validating equilibrium codes.

Physics-based control-oriented modeling of the current density profile evolution in NSTX-Upgrade

Active control of the toroidal current density profile is among those plasma control milestones that the National Spherical Torus eXperiment-Upgrade (NSTX-U) program must achieve to realize its next-step operational goals. Motivated by the coupled, nonlinear, multivariable, distributed-parameter plasma dynamics, the first step towards control design is the development of a physics-based, control-oriented model for the current profile evolution in response to non-inductive current drives and heating systems. The evolution of the toroidal current density profile is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The proposed control-oriented model predicts the spatial-temporal evolution of the current density profile by combining the nonlinear MDE with physics-based correlations obtained at NSTX-U for the electron density, electron temperature, and non-inductive current drives (neutral beams). The resulting first-principles-driven, control-oriented model is tailored for NSTX-U based on predictions by the time-dependent transport code TRANSP. Main objectives and possible challenges associated with the use of the developed model for the design of both feedforward and feedback controllers are also discussed.

Model-based optimal scenario planning in EAST

Ongoing work in the fusion community focuses on developing advanced plasma scenarios characterized by high plasma confinement, magnetohydrodynamic (MHD) stability, and noninductively driven plasma current. The toroidal current density profile, or alternatively the q profile, together with the normalized beta, are often used to characterize these advanced scenarios. The development of these advanced scenarios is experimentally carried out by specifying the devices’ actuator trajectory waveforms, such as the total plasma current, the plasma density, and the auxiliary heating and current-drive (H&CD) sources based on trial-and-error basis. In this work, a model-based numerical optimization approach is followed to complement the experimental effort on actuator trajectory planning in the EAST tokamak. The evolution of the q profile is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). In this work, the MDE is combined with physics-based correlations obtained from EAST experimental data for the plasma density, temperature, resistivity and non-inductive current drives to develop a control-oriented nonlinear PDE model. The optimization objective is to design feedforward trajectories for the plasma current, plasma density, electron cyclotron heating power, neutral beam injection power and lower hybrid current drive power that steer the plasma to desired q profile and βN such that the achieved state is stationary in time. The optimization is subject to the plasma dynamics (described by the physics-based PDE model) and plasma state and actuator constraints, such as the maximum available amount of H&CD power and MHD stability limits. This defines a nonlinear, constrained optimization problem that is solved by employing sequential quadratic programming. The optimized actuator trajectories are assessed in nonlinear transport simulations in preparation for experimental tests in EAST.

EAST equilibrium current profile reconstruction using polarimeter-interferometer internal measurement constraints

The first equilibrium reconstruction of EAST current-density profile based on internal Faraday rotation measurements provided by the POlarimeter-INTerferometer (POINT) diagnostic is demonstrated using the EFIT equilibrium reconstruction code. EFIT incorporates 11 simultaneous line-integrated density and Faraday effect measurements from POINT to self-consistently reconstruct the equilibrium toroidal current density profile using a Faraday rotation reconstruction algorithm. It is shown that the POINT measurements can be applied to improve the accuracy of core plasma current density and q profile on EAST. Comparisons of magnetic surfaces and the q profile reconstructed using external magnetic data against those using magnetic and POINT data are presented. Equilibrium reconstructions using POINT data are found to be consistent with sawtooth phenomena. The sensitivity of equilibrium reconstruction to POINT measurements indicates Faraday rotation provides important constraints for determining the current profile.

Preliminary conceptual design of a medium sized tokamak (IST-1)

In this paper an attempt is made to estimate the main parameters of the Iranian superconducting tokamak as a medium sized tokamak. In the first stage, the production and confinement of ohmically heated plasma is considered. Considering the aim of the design and the kink stability limit, three main parameters are assumed to be known. Using the known theoretical, empirical scale laws and numerical solution of Grad–Shafranov equation for a D-shaped plasmas and also considering the correction terms due to triangularity of the torus cross section, other physical and geometrical parameters have been estimated. The magnetic flux surfaces, plasma pressure and toroidal current density profiles are found by solving of Grad–Shafranov equation as an eigenvalue problem using finite element method. The preliminary results are compared with some recent tokamaks now in operation in different research centers.

MHD Equilibrium with Reversed Current Density and Magnetic Islands Revisited: the Vacuum Vector Potential Calculus

The solution of Grad-Shafranov equation determines the stationary behavior of fusion plasma inside a tokamak. To solve the equation it is necessary to know the toroidal current density profile. Recent works show that it is possible to determine a magnetohydrodynamic (MHD) equilibrium with reversed current density (RCD) profiles that presents magnetic islands. In this work we show analytical MHD equilibrium with a RCD profile and analyze the structure of the vacuum vector potential associated with these equilibria using the virtual casing principle.

Toroidal current profile control during low confinement mode plasma discharges in DIII-D via first-principles-driven model-based robust control synthesis

In order for ITER to be capable of operating in advanced tokamak operating regimes, characterized by a high fusion gain, good plasma confinement, magnetohydrodynamic stability and a non-inductively driven plasma current, for extended periods of time, several challenging plasma control problems still need to be solved. Setting up a suitable toroidal current density profile in the tokamak is key for one possible advanced operating scenario characterized by non-inductive sustainment of the plasma current. At the DIII-D tokamak, the goal is to create the desired current profile during the ramp-up and early flat-top phases of the plasma discharge and then actively maintain this target profile for the remainder of the discharge. The evolution in time of the toroidal current profile in tokamaks is related to the evolution of the poloidal magnetic flux profile, which is modelled in normalized cylindrical coordinates using a first-principles, nonlinear, dynamic partial differential equation (PDE) referred to as the magnetic diffusion equation. The magnetic diffusion equation is combined with empirical correlations developed from physical observations and experimental data from DIII-D for the electron temperature, the plasma resistivity and the non-inductive current drive to develop a simplified, control-oriented, nonlinear, dynamic PDE model of the poloidal flux profile evolution valid for low confinement mode discharges. In this work, we synthesize a robust feedback controller to reject disturbances and track a desired reference trajectory of the poloidal magnetic flux gradient profile by employing the control-oriented model of the system. A singular value decomposition of the static gain matrix of the plant model is utilized to identify the most relevant control channels and is combined with the dynamic response of system around a given operating trajectory to design the feedback controller. A general framework for real-time feedforward + feedback control of magnetic and kinetic plasma profiles was implemented in the DIII-D Plasma Control System and was used to demonstrate the ability of the feedback controller to control the toroidal current profile evolution in the DIII-D tokamak. These experiments constitute the first time ever a first-principles-driven, model-based, closed-loop magnetic profile controller was successfully implemented and tested in a tokamak device.

Identification of edge-localized moments of the current density profile in a tokamak equilibrium from external magnetic measurements

The flux surface topology of a toroidal plasma bounded by a magnetic separatrix allows edge moments of the toroidal current density profile to be identified in an MHD equilibrium reconstruction code using only external magnetic measurements. This is demonstrated analytically for simple plasma shapes and applied to experimental data on the ASDEX Upgrade tokamak where CLISTE reconstructions from magnetic data are shown to be consistent with those obtained from a more complete set of diagnostic data. An independent demonstration of edge current profile recoverability is obtained by analysing the reconstruction errors for a database of Monte Carlo-generated equilibria.

MHD Instabilities in Current Carrying Heliotron Plasmas

Instabilities in low beta l = 2 heliotron plasmas with peaked toroidal current density profiles are investigated using resistive reduced magnetohydrodynamic equations. Such heliotron plasmas can have a nonmonotonic rotational transform ῑ profile with two ῑ = 2/3 rational surfaces. When the distance between the resonant surfaces is large, resistive instabilities can be found. Current-driven ideal modes with larger growth rates appear when the minimum of the rotational transform becomes just above the rational number 2/3 and there is no resonant surface. The existence of this non-resonant mode is explained by the expression for the current-driven term of the plasma potential energy.

Internal magnetic field structure and parallel electric field profile evolution during the sawtooth cycle in MST

Temporal dynamics of the magnetic field, parallel current density, parallel electric field and safety factor profiles in the core of a high-temperature reversed-field pinch plasma are experimentally resolved. Measurements are realized using a high-speed polarimeter–interferometer diagnostic which is employed to simultaneously determine the electron density, toroidal current density and poloidal magnetic field profiles. Combined with external magnetic measurements, these data allow determination of the equilibrium profile dynamics during individual sawtooth magnetic relaxation events. At the sawtooth crash, the E∥ profile has a large positive peak on-axis, and a negative peak of slightly smaller value near the reversal surface demonstrating the need for a dynamo to sustain the plasma equilibrium.

Stability of highly shifted equilibria in a large aspect ratio low-field tokamak

In the long run, the economics of fusion will dictate that reactors confine large plasma pressure rather efficiently. A possible route manifests itself as equilibria with large shift of the plasma magnetic axis. This shift compresses the flux surfaces on the outer part of the plasma, hereby increasing the allowable plasma pressure a machine can confine for a given toroidal magnetic field, which is the main cost of the device. As a first step toward a reactor, we propose investigating the stability of such configurations in a low magnetic field high aspect ratio machine. By focusing our arguments solely on the shape of the toroidal plasma current density profile we discuss the stability of highly shifted equilibria and their robustness to current profile variations that could occur in actual experiments. The evolution of the plasma parameters, as the beta poloidal is increased, is also examined to give a better understanding of the difference in performance between the various regimes.

Mechanisms of disruptions caused by noble gas injection into tokamak plasmas

Noble gas injection for disruption mitigation in DIII-D is simulated. The simulation of the first two stages of the disruption is performed: the first one is the neutral gas jet penetration through the background plasmas, and the second one is the instability growth. In order to simulate the first stage, the MHD pellet code LLP with improved radiation model for noble gas is used. Plasma cooling at this stage is provided by the energy exchange with the jet. The opacity effects in radiation losses are found to be important in the energy balance calculations. The magnetic surfaces in contact with the jet are cooled significantly; however, the temperature as well as the electric conductivity, remains high. The cooling front propagates towards the plasma centre. It has been shown that the cooling front is accompanied by strongly localized ‘shark fin-like’ perturbation in toroidal current density profile. The simplified cylindrical model shows that the cooling front is able to produce the internal kink-like mode with growth rate significantly higher than the tearing mode. The unstable kink perturbation obtained is non-resonant for any magnetic surface, both inside the plasma column, and in the vacuum space outside the separatrix. The mode disturbs mainly the core region. The growth time of the ‘shark fin-like’ mode is higher than the Alfven time by a factor of 10–100 for DIII-D parameters.

Exact magnetohydrodynamic equilibria with flow and effects on the Shafranov shift

Exact solutions of the equation governing the equilibrium magnetohydrodynamic states of an axisymmetric plasma with incompressible flows of arbitrary direction [Phys. Pasmas 5 (1998) 2378] are constructed for toroidal current density profiles peaked on the magnetic axis in connection with the ansatz S=− ku, where S= d/ du[ ϱ( dΦ/ du) 2] ( k is a parameter, u labels the magnetic surfaces; ϱ( u) and Φ( u) are the density and the electrostatic potential, respectively). They pertain to either unbounded plasmas of astrophysical concern or bounded plasmas of arbitrary aspect ratio. For k=0, a case which includes flows parallel to the magnetic field, the solutions are expressed in terms of Kummer functions while for k≠0 in terms of Airy functions. On the basis of a tokamak solution with k≠0 describing a plasma surrounded by a perfectly conducted boundary of rectangular cross-section it turns out that the Shafranov shift is a decreasing function which can vanish for a positive value of k. This value is larger the smaller the aspect ratio of the configuration.

Tokamak equilibria with reversed current density.

Observations of nearly zero toroidal current in the central region of tokamaks (the "current hole") raises the question of the existence of toroidal equilibria with very low or reversed current in the core. The solutions of the Grad-Shafranov equilibrium equation with hollow toroidal current density profile including negative current density in the plasma center are investigated. Solutions of the corresponding eigenvalue problem provide simple examples of such equilibrium configurations. More realistic equilibria with toroidal current density reversal are computed using a new equilibrium problem formulation and computational algorithm which do not assume nested magnetic surfaces.