Poloidal Flux Function Research Articles

Bayesian inference of axisymmetric plasma equilibrium

We present a Bayesian method for inferring axisymmetric plasma equilibria from the magnetic field and plasma pressure measurements. The method calculates all possible solutions for plasma current and pressure distributions consistent with the measurements and magnetohydrodynamic (MHD) force balance. Toroidal plasma current and magnetic field coils are modelled as a set of axisymmetric current-carrying solid beams. The other parameters such as plasma pressure and poloidal current flux are given as a function of poloidal magnetic flux, which is determined given a 2D current distribution. Plasma pressure and poloidal current flux profiles are modelled as Gaussian processes whose smoothness is optimally chosen based on the principle of Occam’s razor. To find equilibrium solutions, we introduce an MHD force balance constraint at every plasma current beam as a part of the prior knowledge. Given all these physical quantities, predictions calculated by the predictive (forward) models for diagnostics are compared to the observations. The high dimensional complex posterior probability distribution is explored by a new algorithm based on the Gibbs sampling scheme.

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component BΨP of the magnetic field vector with respect to the poloidal flux function ΨP. In an unperturbed plasma, BΨP can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of BΨP is small. Here, we are interested in the question whether Hamiltonian conservation laws are still satisfied when BΨP is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without BΨP yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with BΨP retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

Neural network tokamak equilibria with incompressible flows

We present several numerical solutions to a generalized Grad–Shafranov equation (GGSE), which governs axisymmetric plasma equilibria with incompressible flows of arbitrary direction, using fully connected, feed-forward, deep neural networks, also known as multi-layer perceptrons. Such artificial neural networks (ANNs) are trained to approximate tokamak-relevant equilibria upon minimizing the GGSE mean squared residual in the plasma volume and the poloidal flux function on the plasma boundary. Solutions for the Solovev and the general linearizing ansatz for the free functions involved in the GGSE are obtained and benchmarked against known analytic solutions. We also construct a nonlinear equilibrium incorporating characteristics relevant to the high confinement mode. In our numerical experiments, it was observed that changing the radial distribution of the training points has a surprisingly small effect on the accuracy of the trained solution. In particular, it is shown that localizing the training points at the plasma edge results in ANN solutions that describe quite accurately the entire magnetic configuration, thus demonstrating the interpolation capabilities of the ANNs.

Magnetic Force-Free Theory: Nonlinear Case

In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem ∇×B=ΛB with some field conditions B∂Ω on the boundary ∂Ω of the plasma region. In many physical situations, it has been noticed that Λ is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Λ=Λ(ψ) with ψ being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of ψ with respect to a small parameter α, is developed. The Grad–Shafranov equation for ψ is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of spheromaks or protosphera structure of the plasma is determined in the case of nonconstant Λ.

Green function for the Grad-Shafranov operator

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

A framework for the assessment and control of ITER main chamber heat loads

A control-oriented approach, based on real-time equilibrium reconstruction, has been developed to monitor power fluxes to ITER plasma-facing components (PFC). The model describes the deposited heat flux as a poloidal flux function with two main input parameters: the power exhausted across the plasma boundary and the scrape-off layer heat flux width. A module containing weighting factors accounts for the real PFC 3D geometry. These factors are obtained using a new sophisticated GUI interface, SMITER, hosting a magnetic field line tracing code permitting the import and appropriate meshing of full PFC CAD descriptions. This new methodology has been used to examine two critically important first wall (FW) power loading issues for ITER: the start-up phase at low current and power, but requiring direct contact with the inboard wall, and the case of an upward plasma shift in a diverted configuration under high power, high current H-mode burning plasma conditions, which imposes very high power loads on the upper chamber FW panels.

Deep neural network Grad–Shafranov solver constrained with measured magnetic signals

A neural network solving the Grad–Shafranov equation constrained with measured magnetic signals to reconstruct magnetic equilibria in real time is developed. The database created to optimize the neural network’s free parameters contains off-line EFIT results as the output of the network from 1118 KSTAR experimental discharges of two different campaigns. Input data to the network constitute magnetic signals measured by a Rogowski coil (plasma current), magnetic pick-up coils (normal and tangential components of magnetic fields) and flux loops (poloidal magnetic fluxes). The developed neural networks fully reconstruct not only the poloidal flux function but also the toroidal current density function with the off-line EFIT quality. To preserve the robustness of the networks against missing input data, an imputation scheme is utilized to eliminate the required additional training sets with a large number of possible combinations of the missing inputs.

Experimental implementation of a real-time power flux estimator for the ITER first wall on the TCV tokamak

A control-oriented approach to the monitoring of wall power flux densities on ITER has been successfully developed. It is based on real-time equilibrium reconstruction in 2-D which is then used to describe the deposited heat flux as a poloidal flux function with user specified parameters for the power exhausted into the scrape-off layer (SOL) and the SOL heat flux width. To account for the real 3-D geometry of the plasma-facing components (PFC), appropriate weighting factors are derived from magnetic field line tracing in 3-D. Integration of the 3-D effect is performed with a new GUI-based software environment, SMITER, incorporating a field line tracer and permitting import and meshing of PFC CAD models. The paper discusses the experimental demonstration of the model-based wall heat flux algorithm on the TCV tokamak, reporting on the benchmarking of the new code package, SMITER against infra-red camera heat flux measurements and the derivation of the component shaping weighting factors. A comparison of the real-time estimation of the peak power flux and its spatial location against the off-line infra-red measurement for limiter plasma configurations is presented.

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.

A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries

Existing finite element implementations for the computation of free-boundary axisymmetric plasma equilibria approximate the unknown poloidal flux function by standard lowest order continuous finite elements with discontinuous gradients. As a consequence, the location of critical points of the poloidal flux, that are of paramount importance in tokamak engineering, is constrained to nodes of the mesh leading to undesired jumps in transient problems. Moreover, recent numerical results for the self-consistent coupling of equilibrium with resistive diffusion and transport suggest the necessity of higher regularity when approximating the flux map. In this work we propose a mortar element method that employs two overlapping meshes. One mesh with Cartesian quadrilaterals covers the vacuum chamber domain accessible by the plasma and one mesh with triangles discretizes the region outside. The two meshes overlap in a narrow region. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details outside this region. The continuity of the numerical solution in the region of overlap is weakly enforced by a mortar-like mapping.

Fixed boundary toroidal plasma equilibria with toroidal flows

The fixed boundary toroidal plasma equilibria with toroidal flows are investigated by solving the modified Grad-Shafranov equation numerically in the cylindrical coordinate system. For normal equilibrium configurations with geometry and profiles similar to usual tokamaks with no flow, it is found that the effect of flow is to lead to an outward shift of the magnetic flux surfaces, together with the profiles of pressure, and mass and current densities. The shifts could become significant when the toroidal flow Mach number exceeds 0.5. For non-conventional current profiles, even for the usual tokamak geometry, novel current reversal equilibrium configurations may result, sometimes with changed topology in the poloidal flux function. This change in the topology of plasma equilibrium can be attributed to the large toroidal flow. The computed results may correspond to situations of intense tangential injection during the low toroidal current phase in expected experimental situations.

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.

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria

We present ECOM (Equilibrium solver via COnformal Mapping), a fast and accurate fixed boundary solver for toroidally axisymmetric magnetohydrodynamic equilibria with or without a toroidal flow. ECOM combines conformal mapping and Fourier and integral equation methods on the unit disk to achieve exponential convergence for the poloidal flux function as well as its first and second partial derivatives. As a consequence of its high order accuracy, for dense grids and elongations comparable to or smaller than the elongation of ITER, ECOM computes key quantities such as the safety factor and the magnetic shear with higher accuracy than the finite element based code CHEASE (Lütjens et al., 1996) at equal run time. ECOM has been developed to provide equilibrium quantities and details of the flux contour geometry as inputs to stability, wave propagation and transport codes.

A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θBφ=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θBφ=aA|A|1/n, n a positive constant, admits separable fields with A=An(θ)rn, posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and An(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=H(θ,φ)rn+2 promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as discussed in the Appendix.

CENTORI: A global toroidal electromagnetic two-fluid plasma turbulence code

A new global two-fluid electromagnetic turbulence code, CENTORI, has been developed for the purpose of studying magnetically-confined fusion plasmas on energy confinement timescales. This code is used to evolve the combined system of electron and ion fluid equations and Maxwell equations in toroidal configurations with axisymmetric equilibria. Uniquely, the equilibrium is co-evolved with the turbulence, and is thus modified by it. CENTORI is applicable to tokamaks of arbitrary aspect ratio and high plasma beta. A predictor–corrector, semi-implicit finite difference scheme is used to compute the time evolution of fluid quantities and fields. Vector operations and the evaluation of flux surface averages are speeded up by choosing the Jacobian of the transformation from laboratory to plasma coordinates to be a function of the equilibrium poloidal magnetic flux. A subroutine, GRASS, is used to co-evolve the plasma equilibrium by computing the steady-state solutions of a diffusion equation with a pseudo-time derivative. The code is written in Fortran 95 and is efficiently parallelised using Message Passing Interface (MPI). Illustrative examples of output from simulations of a tearing mode in a large aspect ratio tokamak plasma and of turbulence in an elongated conventional aspect ratio tokamak plasma are provided.

Analytical solutions for Tokamak equilibria with reversed toroidal current

In tokamaks, an advanced plasma confinement regime has been investigated with a central hollow electric current with negative density which gives rise to non-nested magnetic surfaces. We present analytical solutions for the magnetohydrodynamic equilibria of this regime in terms of non-orthogonal toroidal polar coordinates. These solutions are obtained for large aspect ratio tokamaks and they are valid for any kind of reversed hollow current density profiles. The zero order solution of the poloidal magnetic flux function describes nested toroidal magnetic surfaces with a magnetic axis displaced due to the toroidal geometry. The first order correction introduces a poloidal field asymmetry and, consequently, magnetic islands arise around the zero order surface with null poloidal magnetic flux gradient. An analytic expression for the magnetic island width is deduced in terms of the equilibrium parameters. We give examples of the equilibrium plasma profiles and islands obtained for a class of current density profile.

Local properties of the magnetic field in a snowflake divertor

The power-law series for the poloidal magnetic flux function, up to the third-order terms, is presented for the case where two nulls of the poloidal magnetic field are separated by a small distance, as in a snowflake divertor. Distinct from the earlier results, no assumptions about the field symmetry are made. Conditions for the realization of an exact snowflake are expressed in terms of the coefficients of the power series. It is shown that, by a proper choice of the coordinate frame in the poloidal plane, one can obtain efficient similarity solutions for the separatrices and flux surfaces in the divertor region: the whole variety of flux surface shapes can be characterized by a single dimensionless parameter. Transition from a snowflake-minus to a snowflake-plus configuration in the case of no particular symmetry is described. The effect of the finite toroidal current density in the divertor region is assessed for the case of no particular symmetry.

On the accuracy of the symmetric ergodic magnetic limiter map in tokamaks

AbstractA new symmetric symplectic map for an ergodic magnetic limiter (EML) is proposed. A rigorous mapping technique based on the Hamilton–Jacobi equation is used for its derivation. The system is composed of the equilibrium field, which is fully integrable, and a Hamiltonian perturbation. The equilibrium poloidal flux function is a solution of the Grad–Schlüter–Shafranov equation. This equation is written in polar toroidal coordinate in order to take into account the outward Shafranov shift. The static perturbation field breaks the exact axisymmetry of the equilibrium field and creates a region of chaotic field lines near the plasma edge. The new symmetric EML map is compared with the conventional (asymmetric) EML map which is derived by applying delta-function method. The accuracy of the maps is considered through mean energy error criterion and maximal Lyapunov exponents. For asymmetric and symmetric maps the approximate location of the main cantorus near the edge of plasma is determined with high accuracy by using mean energy error. The forward–backward error criterion is applied to show the relation between the accuracy of the symmetric EML map and the number of EML rings. We also report on the effect of the number of EML rings on the maximal Lyapunov exponent of the symmetric EML map.

Structure of magnetic fields in non-convective stars

(Abridged) We develop a theoretical framework to construct axisymmetric magnetic equilibria in stars, consisting of both poloidal and toroidal magnetic field components. In a stationary axisymmetric configuration, the poloidal current is a function of the poloidal magnetic flux only, and thus should vanish on field lines extending outside of the star. Non-zero poloidal current is limited to a set of toroid-shape flux surfaces fully enclosed inside the star. If we demand that there are no current sheets, then on the separatrix delineating the regions of zero and finite toroidal magnetic field both the poloidal flux function and its derivative should match. Thus, for a given magnetic field in the bulk of the star, the elliptical Grad-Shafranov equation that describes magnetic field structure inside the toroid is an ill-posed problem, with both Dirichlet and Newman boundary conditions and {\it a priori} unknown distribution of toroidal and poloidal electric currents. We discuss a procedure which allows to solve this ill-posed problem by adjusting the unknown current functions. We find a poloidal current-carrying solution that leaves the shape of the flux function and, correspondingly, the toroidal component of the electric current the same as in the case of no poloidal current. The equilibria discussed in this paper may have arbitrary large toroidal magnetic field, and may include a set of stable equilibria. The method developed here can also be applied to magnetic structure of differentially rotating stars, as well as to calculate velocity field in incompressible isolated fluid vortex with a swirl.

Hall equilibrium of thin Keplerian discs embedded in mixed poloidal and toroidal magnetic fields

Axisymmetric steady-state weakly ionized Hall-MHD Keplerian thin disks are investigated by using asymptotic expansions in the small disk aspect ratio \epsilon. The model incorporates the azimuthal and poloidal components of the magnetic fields in the leading order in \epsilon. The disk structure is described by an appropriate Grad-Shafranov equation for the poloidal flux function \psi that involves two arbitrary functions of \psi for the toroidal and poloidal currents. The flux function is symmetric about the midplane and satisfies certain boundary conditions at the near-horizontal disk edges. The boundary conditions model the combined effect of the primordial as well as the dipole-like magnetic fields. An analytical solution for the Hall equilibrium is achieved by further expanding the relevant equations in an additional small parameter \delta that is inversely proportional to the Hall parameter. It is thus found that the Hall equilibrium disks fall into two types: Keplerian disks with (i) small (Rd ~\delta^0) and (ii) large (Rd > \delta^k, k > 0) radius of the disk. The numerical examples that are presented demonstrate the richness and great variety of magnetic and density configurations that may be achieved under the Hall-MHD equilibrium.