Equilibrium reconstruction for Single Helical Axis reversed field pinch plasmas
Emilio Martines, Rita Lorenzini, Barbara Momo, David Terranova, Paolo Zanca, Alberto Alfier, Federica Bonomo, Alessandra Canton, Alessandro Fassina, Paolo Franz, Paolo Innocente
Single Helical Axis (SHAx) configurations are emerging as the natural state for high current reversed field pinch (RFP) plasmas. These states feature the presence of transport barriers in the core plasma. Here we present a method for computing the equilibrium magnetic surfaces for these states in the force-free approximation, which has been implemented in the SHEq code. The method is based on the superposition of a zeroth order axisymmetric equilibrium and of a first order helical perturbation computed according to Newcomb's equation supplemented with edge magnetic field measurements. The mapping of the measured electron temperature profiles, soft X-ray emission and interferometric density measurements on the computed magnetic surfaces demonstrates the quality of the equilibrium reconstruction. The procedure for computing flux surface averages is illustrated, and applied to the evaluation of the thermal conductivity profile. The consistency of the evaluated equilibria with Ohm's law is also discussed.
In this paper a new idea is presented which is able to predict the scaling law with the aspect ratio for the toroidal mode number of the so called Single Helicity states in Reversed Field Pinch plasmas. The model starts considering the pure electromagnetic interaction of an helical current with a conductive toroidal shell. The stability properties of the plasma become then essential in determining the emergence of the observed dominant mode.
In the reversed field pinch RFX-mod strong electron temperature gradients develop when the Single-Helical-Axis regime is achieved. Gyrokinetic calculations show that in the region of the strong temperature gradients microtearing instabilities are the dominant turbulent mechanism acting on the ion Larmor radius scale. The quasi-linear evaluation of the electron thermal conductivity is in good agreement with the experimental estimates.
The presence of impurities is considered in gyrokinetic calculations of ion temperature gradient (ITG) instabilities and turbulence in the reversed field pinch device RFX-mod. This device usually exhibits hollow Carbon/Oxygen profiles, peaked in the outer core region. We describe the role of the impurities in ITG mode destabilization, and analyze whether ITG turbulence is compatible with their experimental gradients.
An analysis of experimental data from the RFX (Reversed Field eXperiment) reversed field pinch machine [L. Fellin, P. Kusstatscher and G. Rostagni, Fusion Eng. Des. {\bf 25}, 315 (1995)] is carried out to investigate the possible existence of deterministic chaos in the edge plasma region. The mathematical tools used include Lyapunov exponents, Kaplan-Yorke dimension, minimum embedding dimension estimates and nonlinear forecasting. The whole analysis agrees in ruling out the possibility of low-dimensional chaos: The dimension of the underlying dynamical system is estimated to be $ > 10$. From a…
The kinetic effects, due to the mode resonance with thermal particle drift motions in the reversed field pinch (RFP) plasmas, are numerically investigated for the stability of the resistive wall mode, using a non-perturbative MHD-kinetic hybrid formulation. The kinetic effects are generally found too weak to substantially change the mode growth rate, or the stability margin, re-enforcing the fact that the ideal MHD model is rather adequate for describing the RWM physics in RFP experiments.
First direct experimental evidence of high frequency, high toroidal mode number (n>20), magnetic fluctuations due to unstable resistive interchange modes (g-modes) resonant in the edge region of a reversed field pinch (RFP) plasma is presented. Experimental characterization of time and space periodicities of the modes is provided by means of highly resolved in-vessel edge and insertable magnetic diagnostics. It is found that the spectral mode properties are in good agreement with the predictions of the theoretical linear resistive magnetohydrodynamic stability analysis. A simple model is propo…
We show that the self-organized single-helical-axis (SHAx) and double-helical-axis (DAx) states in reversed field pinches can be reproduced in a minimally constrained equilibrium model using only five parameters. This is a significant reduction on previous representations of the SHAx which have required an infinite number of constraints. The DAx state, which has a non-trivial topology, has not been previously reproduced using an equilibrium model that preserves this topological structure. We show that both states are a consequence of transport barrier formation in the plasma core, in agreement…
A recent study [1] about the scaling with the aspect ratio (i.e. the ratio between the major and minor radius of the torus) of the dominant mode in Single Helical (SH) Reverse Field Pinch (RFP) plasmas has shown, at intermediate aspect ratio, that the dominant toroidal mode number in the helical states can be interpreted as the result of a relaxation process. In this work, the theoretical model is compared and validated with the experimental data obtained in the low aspect ratio RELAX device [2]. [1] R. Paccagnella, Phys. of Plasmas 23 (2016) 092512. [2] S. Masamune et al, J. Phys. Soc. Japan …
Till now the magnetohydrodynamic (MHD) simulation of the reversed field pinch (RFP) has been performed by assuming axis-symmetric radial time independent dissipation profiles. In helical states this assumption is not correct since these dissipations should be flux functions, and should exhibit a helical symmetry as well. Therefore more correct simulations should incorporate self-consistent dissipation profiles. As a first step in this direction, the case of uniform dissipation profiles was considered by using the 3D nonlinear visco-resistive MHD code SpeCyl. It is found that a flattening of th…
Transport of field lines is studied for a realistic model of magnetic field configuration in a Reversed Field Pinch. It is shown that transport is anomalous, i.e., it cannot be described within the standard diffusive paradigm. To fit numerical results we present a transport model based upon the Continuous Time Random Walk formalism. Fairly good quantitative agreement appears for exponential memory functions.
Two-fluid boundary turbulence simulations in reversed field pinch plasmas
University of Padua（帕多瓦大学）；Consorzio RFX (CNR, ENEA, INFN, Università di Padova, Acciaierie Venete SpA)（RFX联盟（意大利国家研究委员会、国家新能源研究中心、意大利国家核物理研究所、帕多瓦大学、威尼斯钢铁公司））；Istituto per la Scienza e la Tecnologia dei Plasmi (ISTP), CNR（等离子体科学与技术研究所，意大利国家研究委员会）
In this paper a relaxation theory, proposed several years ago for plasmas where a single dominant mode is present, is revisited. The solutions of the related eigenvalue problem are numerically calculated and discussed. Although these solutions are very rich and can reproduce well the magnetic fields measured in experiments, there is no manner within the theory to determine the dominant helicity, which is a free parameter in the model. To find the preferred helicity in the system, we propose to minimize the distance of the relaxed state from another state which is constructed as a two region ge…
A new method to obtain the radial profile of the magnetic perturbation in a toroidal force-free plasma having circular cross section is developed. The method is quite general and can be applied to any circular low-beta plasma once suitable boundary conditions are imposed. In particular, due to its characteristics, it is appropriate for describing the Reversed Field Pinch (RFP) configuration. The curvilinear metric of the toroidal geometry is described by adopting flux co-ordinates, and the magnetic field is expressed by the contravariant representation with two independent potentials. The meth…
The drift wave in the presence of impurity ions was investigated numerically in reversed field pinch (RFP) plasmas, using the gyrokinetic integral eigenmode equation. It was found that in RFP plasmas with hollow density profiles, an increase in $k_θρ_s$ increases the growth rate of the ion temperature gradient (ITG). Comparing the results of regular and hollow plasma density profile shows that the ITG mode under the hollow density profile is much harder to be excited. For the impurities' effects, when the impurities' density gradient is opposite to the primary ions, namely when $L_{ez}$ is neg…
A new non-disruptive error field (EF) assessment technique not restricted to low density and thus low beta was demonstrated at the EXTRAP-T2R reversed field pinch. Stable and marginally stable external kink modes of toroidal mode number n=10 and n=8, respectively, were generated, and their rotation sustained, by means of rotating magnetic perturbations of the same n. Due to finite EFs, and in spite of the applied perturbations rotating uniformly and having constant amplitude, the kink modes were observed to rotate non-uniformly and be modulated in amplitude. This behavior was used to precisely…
The Multi-region Relaxed MHD (MRxMHD) has been successful in the construction of equilibria in three-dimensional (3D) configurations. In MRxMHD, the plasma is sliced into sub-volumes separated by ideal interfaces, each undergoing relaxation, allowing the formation of islands and chaos. The resulting equilibrium has a stepped pressure profile across sub-volumes. The Stepped Pressure Equilibrium Code (SPEC) [S.R. Hudson et al., Phys. Plasmas 19, 112502 (2012)] was developed to calculate MRxMHD equilibria numerically. In this work, we have extended the SPEC code to compute MRxMHD equilibria with …
Axisymmetric relaxed states of a cylindrical plasma column are found analytically in both standard and Hall magnetohydrodynamics (MHD) by complete minimization of energy with constraints imposed by invariants inherent in corresponding models. It is shown that the relaxed state in Hall MHD is the force-free magnetic field with uniform axial flow and/or rigid azimuthal rotation. The relaxed states in standard MHD are more complex due to the coupling between velocity and magnetic field. Application of these states for reversed-field pinches (RFP) is discussed.
Turbulence induced by the ion temperature gradient (ITG) is investigated in the helical and axisymmetric plasma states of a reversed field pinch device by means of gyrokinetic calculations. The two magnetic configurations are systematically compared, both linearly and nonlinearly, in order to evaluate the impact of the geometry on the instability and its ensuing transport, as well as on the production of zonal flows. Despite its enhanced confinement, the high-current helical state demonstrates a lower ITG stability threshold compared to the axisymmetric state, and ITG turbulence is expected to…
The drift waves in field-reversed configurations without a toroidal magnetic field, therefore no shear, play an important role in plasma transport. The short connection length of the poloidal field in these systems leads to significant stabilization by influencing the wave particle resonance. The field reversed configuration is modeled by the cylindrical Bennett pinch in the limit of large aspect ratio. The radial eigenmode equation for the universal mode is derived from kinetic theory and the method of quadratic forms is used to study its stability. The short connection lengths of the field l…
Advanced neutral particles analyzer for plasma diagnostic with possibility of simultaneous measurements of energy distributions of D and H ions has developed in the Budker Institute of Nuclear Physics. The analyzer was used in two plasma facilities with injection of fast neutrals - on the MST reversed field pinch (University of Wisconsin) and the field reversed configuration C-2 (Tri Alpha Energy). In this paper, the design of the analyzer, calculation of efficiency of registration, results of analyzer calibration and experimental results from MST and C-2 experiments are presented
Single Helical Axis (SHAx) states obtained in high current reversed field pinch (RFP) plasmas display, aside from a dominant mode in the m=1 spectrum, also a dominant m=0 mode, with the same toroidal mode number as the m=1 one. The two modes have a fixed phase relationship. The island chain created by the m=0 mode across the reversal surface gives rise, at shallow reversal of the toroidal field, to an X-point structure which separates the last closed flux surface from the first wall, creating a divertor-like configuration. The plasma-wall interaction is found to be related to the connection le…
Achieving impurity and helium ash control is a crucial issue in the path towards fusion-grade magnetic confinement devices, and this is particularly the case of helical reactors, whose low-collisionality ion-root operation scenarios usually display a negative radial electric field which is expected to cause inwards impurity pinch. In these work we discuss, based on experimental measurements and standard predictions of neoclassical theory, how plasmas of very low ion collisionality, similar to those observed in the impurity hole of the Large Helical Device, can be an exception to this general r…
The interaction of the two counter-propagating ultrashort laser pulses with a singular wavefronts in the thin slice of the underdense plasma is considered. It is shown that ion-acoustic wave is excited via Brillouin three-wave resonance by corkscrew interference pattern of a paraxial singular laser beams. The orbital angular momentum carried by light is transferred to plasma ion-acoustic vortex. The rotation of the density perturbations of electron fluid is the cause of helical current which produce the kilogauss axial quasi-static magnetic field. The exact analytical configurations are presen…
The relaxation of a helical magnetic field ${\bf B}({\bf x}, t)$ in a high-conductivity plasma contained in the annulus between two perfectly conducting coaxial cylinders is considered. The plasma is of low density and its pressure is negligible compared with the magnetic pressure; the flow of the plasma is driven by the Lorentz force and and energy is dissipated primarily by the viscosity of the medium. The axial and toroidal fluxes of magnetic field are conserved in the perfect-conductivity limit, as is the mass per unit axial length. The magnetic field relaxes during a rapid initial stage t…
The experimentally measured single helical states in the low aspect ratio (R/a=2) RELAX device have been compared with the predictions of a cylindrical relaxation theory and a substantial deviation in terms of some macroscopic characteristics has been found, in particular for cases corresponding to a large reversal of the edge toroidal field. In this paper we consider the effect of the toroidal geometry by employing the VMEC and the RelaxFit equilibrium codes. We show that the inclusion of toroidicity produces significant corrections to the cylindrical states, especially for the deep reversed …
Present-days Reversed Field Pinches (RFPs) are characterized by quasi-laminar magnetic configurations in their core, whose boundaries feature sharp internal transport barriers, in analogy with tokamaks and stellarators. The abatement of magnetic chaos leads to the reduction of associated particle and heat transport along wandering field lines. At the same time, the growth of steep temperature gradients may trigger drift microinstabilities. In this work we summarize the work recently done in the RFP RFX-mod in order to assess the existence and the impact upon transport of such electrostatic and…
The final state of turbulent magnetic relaxation in a reversed field pinch is well explained by Taylor's hypothesis. However, recent resistive-magnetohydrodynamic simulations of the relaxation of braided solar coronal loops have led to relaxed fields far from the Taylor state, despite the conservation of helicity. We point out the existence of an additional topological invariant in any flux tube with non-zero field: the topological degree of the field line mapping. We conjecture that this constrains the relaxation, explaining why only one of three example simulations reaches the Taylor state.
The effect of magnetic perturbations (MPs) on the helical self-organization of shaped tokamak plasmas is discussed in the framework of the nonlinear 3D MHD model. Numerical simulations performed in toroidal geometry with the \textsc{pixie3d} code [L. Chacón, Phys. Plasmas {\bf 15}, 056103 (2008)] show that $n=1$ MPs significantly affect the spontaneous quasi-periodic sawtoothing activity of such plasmas. In particular, the mitigation of sawtooth oscillations is induced by $m/n=1/1$ and $2/1$ MPs. These numerical findings provide a confirmation of previous circular tokamak simulations, and are …
We investigate the intermittency of magnetic turbulence as measured in Reversed Field Pinch plasmas. We show that the Probability Distribution Functions of magnetic field differences are not scale invariant, that is the wings of these functions are more important at the smallest scales, a classical signature of intermittency. We show that scaling laws appear also in a region very close to the external wall of the confinement device, and we present evidences that the observed intermittency increases moving towards the wall.
Using mean-field theory, we compute the evolution of the magnetic field in a cylinder with outer perfectly conducting boundaries, an imposed axial magnetic and electric field. The thus injected magnetic helicity in the system can be redistributed by magnetic helicity fluxes down the gradient of the local current helicity of the small-scale magnetic field. A weak reversal of the axial magnetic field is found to be a consequence of the magnetic helicity flux in the system. Such fluxes are known to alleviate so-called catastrophic quenching of the α-effect in astrophysical applications. Applicati…
Three-dimensional, time-dependent numerical simulations of the dynamics of magnetic flux ropes are presented. The simulations are targeted towards an experiment previously conducted at CalTech (Bellan, P. M. and J. F. Hansen, Phys. Plasmas, 5, 1991 (1998)) which aimed at simulating Solar prominence eruptions in the laboratory. The plasma dynamics is described by ideal MHD using different models for the evolution of the mass density. Key features of the reported experimental observations like pinching of the current loop, its expansion and distortion into helical shape are reproduced in the num…
A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, Multi-region Relaxed Magnetohydrodynamics (MRxMHD), which models the plasma-magnetic field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by flexible ideal-MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic …
Programmable control of the inductive electric field enables advanced operations of reversed-field pinch (RFP) plasmas in the Madison Symmetric Torus (MST) device and further develops the technical basis for ohmically heated fusion RFP plasmas. MST's poloidal and toroidal magnetic fields ($B_\text{p}$ and $B_\text{t}$) can be sourced by programmable power supplies (PPSs) based on integrated-gate bipolar transistors (IGBT). In order to provide real-time simultaneous control of both $B_\text{p}$ and $B_\text{t}$ circuits, a time-independent integrated model is developed. The actuators considered…
We study the influence of the shape of the plasma container on the dynamics of the Reversed Field Pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear visco-resistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics are affected by shaping. It is shown that independent of the plasma shape, the quantity $β$, comparing the hydrodynamic pressure to the magnetic pressure, decreases for increasing values of the Lundquist number, but the pressure gradient fluctuations rema…
We present a numerical study of charged particle motion in a time-independent magnetic field in cylindrical geometry. The magnetic field model consists of an unperturbed reversed-shear helical part and a perturbation consisting of a superposition of modes. Contrary to most of the previous studies, the particle trajectories are computed by directly solving the full Lorentz force equations of motion in a six-dimensional phase space using a sixth-order, implicit, symplectic Gauss-Legendre method. The level of stochasticity in the particle orbits is diagnosed using averaged, effective Poincare sec…
