Inverse Aspect Ratio Research Articles

Poloidally asymmetric potential formation on plasma boundary in axisymmetric magnetic field

To study the symmetry of electrical potential, we model plasma transport in the edge region of a toroidal device with two spatial dimensions (2D) and three coordinates for velocities (3V) using a Particle-In-Cell (PIC) code. A two-dimensional magnetic field is applied, including poloidal and toroidal components, which are periodic in the poloidal direction. We discover relationships between the magnetic gradient drift and potential formation using PIC simulation, which has not been captured in other numerical models. We find that the inverse aspect ratio influences the asymmetry of the potential in an axisymmetric magnetic configuration.

Unveiling non-flat profiles within magnetic islands in tokamaks

The presence of non-flat profiles on magnetic island is studied for the first time through gyrokinetic simulations alongside a simplified Lagrangian model. We have identified that inside a magnetic island, the non-flatness of density and temperature profiles is controlled by a dimensionless parameter α≡w*ŝϵ/qρ*, which is a function of normalized island width w*=w/a0, magnetic shear ŝ, inverse aspect ratio ϵ=a0/R, safety factor q, and normalized gyroradius ρ*=ρ/a0. The gyroradius ρ* dependence of the control parameter α leads to a species-selective transition of profiles from flat to concave only for electrons having high α∼O(1). The finding elucidates that electron profiles tend to increasingly deviate from the flat state for a larger magnetic island, in contrast to the conventional wisdom.

An analytical model of how the negative triangularity cuts off the access to the second stable region in tokamak plasmas

We present an analytical model to evaluate the triangularity-shaping effects in accessing the second stable region for the ideal ballooning mode. Our results indicate that if the triangularity is sufficiently negative, the path from the first to the second stable region will be closed. The reason is that negative triangularity can weaken the stabilizing effect of the ‘magnetic well’, and even convert the ‘magnetic well’ into a ‘magnetic hill’, which will destabilize the ballooning mode. We also show that the synergistic effects of elongation, inverse aspect ratio, and safety factor can reopen the path to the second stable region. Through a variational approach, we derive an analytical expression of the critical negative triangularity for closing the access to the second stable region. Furthermore, our analysis reveals that in the second ballooning stable regime, the negative triangularity tends to inhibit the emergence of quasi marginally stable discrete Alfvén eigenmodes. These findings provide a quantitative understanding of how the negative triangularity configuration impacts the confinement of tokamak plasmas.

In situ synchrotron X-ray diffraction analysis of ultrasonically assisted microstructure refinement during laser melting process

This work investigates the effect of ultrasonic treatment (UST) on the microstructure of AA4043 alloy during laser melting process. The study utilizes in situ synchrotron X-ray diffraction (SXRD) and postmortem electron backscatter diffraction (EBSD) techniques to examine the ultrasonic grain refinement mechanism. The SXRD diffraction patterns exhibit higher continuity of major diffraction rings in the UST-test pattern, indicating a more refined grain structure. The presence of new diffraction spots in the post-test diffraction pattern with UST suggests the breakdown of epitaxial growth of columnar dendrites during solidification. Thermal profiles determined by analyzing the lattice parameter evolution show a faster cooling rate with UST. The inverse pole figures and aspect ratio distribution from EBSD reveal that the UST samples exhibit a refined, more equiaxed grain structure along the weld centerline. UST sample also develops more fine-sized low angle grain boundaries (LAGBs). These features offer better hot-cracking resistance in aluminum alloys.

Development of Penning ion gauge for in-situ measurement of neutral pressure in VEST

A Penning Ion Gauge (PIG) has been developed to measure the responsive temporal changes of neutral pressure inside the magnetic fusion devices, and successfully operated in Versatile Experiment Spherical Torus (VEST). To check its reliabilities during in-situ operation, feasibility tests are conducted by specifically investigating both the penning discharge mode transition and the effect of magnetic field alignment on the penning discharge. The mode transition of penning discharge is clearly observed, confirming the developed PIG is applicable to the operation regime of VEST. The PIG is also designed to measure discharge current stably during magnetic field angle variation of upto 15∘. In addition, it has been verified that the operation range of PIG can be expanded by increasing inverse aspect ratio of the gauge cell geometry. Finally, the developed PIG is installed in VEST, measuring a millisecond-order temporal changes in neutral pressure within a range of 3×10−5 mbar to 1×10−3 mbar, under the magnetic field angle variation of up to 15∘. Finally, it is successfully measured rapidly evolving neutral pressures resulting from the plasma wall interaction during ohmic discharges in VEST.

Self-generated vortex flows in a tokamak magnetic island with a background flow

We present a gyrokinetic theory of self-generated E × B vortex flows in a magnetic island in a collisionless tokamak plasma with a background vortex flow. We find that the long-term evolution of the self-generated vortex flows can be classified into two regimes by the background vortex flow potential Φ, with an asymptotic criterion given by , where T is temperature, ε is the inverse aspect ratio and r is the radial coordinate. We find that the background vortex flow above the criterion significantly weakens the toroidal precession-induced long-term damping and structure change of the self-generated vortex flows. That is, the finite background vortex flow is beneficial to maintain the self-generated vortex flows, favorable to an internal transport barrier formation. Our result indicates that the island boundary region is a prominent location for triggering the transition to an enhanced confinement state of the magnetic island.

Shaping effects on the geodesic acoustic mode in tokamaks

The geodesic acoustic mode (GAM) is investigated with gyro-kinetic equations in Miller local equilibrium model for shaped tokamak plasmas with an arbitrary elongation κ and a finite triangularity δ. In particular, the effects of triangularity on GAM frequency and damping rate are analyzed both analytically and numerically. The asymptotic analytical and exact numerical results both show that the frequency almost linearly increases with the triangularity but increases relatively more slowly for a negative δ, which agree well with the TCV observation on the trend. The analytical results clearly claim that the triangularity effect strength is dependent on the inverse aspect ratio and Shafranov shift gradient , while the numerical results indicate that the safety factor q also has a significant impact on the triangularity effects. In addition, the damping rate increases rapidly with triangularity when q is not too large and then saturates when δ is above about 0.3.

Drift kinetic theory of the NTM magnetic islands in a finite beta general geometry tokamak plasma

In (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from , where is the trapped ion banana orbit width, to , making it in closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

Marginally stable current sheets in collisionless magnetic reconnection.

Noncollisional current sheets that form during the nonlinear development of spontaneous magnetic reconnection are characterized by a small thickness, of the order of the electron skin depth. They can become unstable to the formation of plasmoids, which allows the magnetic reconnection process to reach high reconnection rates. In this work, we investigate the marginal stability conditions for the development of plasmoids when the forming current sheet is purely collisionless and in the presence of a strong guide field. We analyze the geometry that characterizes the reconnecting current sheet, and what promotes its elongation. Once the reconnecting current sheet is formed, we identify the regimes for which it is plasmoid unstable. Our study shows that plasmoids can be obtained, in this context, from current sheets with an aspect ratio much smaller than in the collisional regime, and that the plasma flow channel of the marginally stable current layers maintains an inverse aspect ratio of 0.1.

Prospects for H-mode inhibition in negative triangularity tokamak reactor plasmas

Instability to high toroidal mode number (n) ballooning modes has been proposed as the primary gradient-limiting mechanism for tokamak equilibria with negative triangularity (NT) (δ) shaping, preventing access to strong H-mode regimes when δ is sufficiently negative. To understand how this mechanism extrapolates to reactor conditions, we model the infinite-n ballooning stability as a function of internal profiles and equilibrium shape using a combination of the CHEASE and BALOO codes. While the critical δ required for avoiding 2nd stability to high-n modes is observed to depend in a complicated way on various shaping parameters, including the equilibrium inverse aspect ratio, elongation and squareness, equilibria with NT are robustly prohibited from accessing the 2nd stability region, offering the prediction that NT reactors should maintain L-mode-like operation. In order to access high-n 2nd stability, the local shear over the entire bad curvature region must be sufficiently negative to overcome curvature destabilization on the low field side. Scalings of the ballooning-limited pedestal height are provided as a function of plasma parameters to aid future scenario design.

Tensile modulus of halloysite-nanotube-based system assuming the defective interfacial bonding between polymer medium and halloysite nanotube

In this paper, essential interfacial shear modulus (Gc) in halloysite nanotubes (HNT) nanocomposites is expressed to define the effective inverse aspect ratio and effective volume portion of HNT in the samples. Additionally, Halpin-Tsai model is advanced by the mentioned terms for the modulus of HNT-filled system assuming the defective bonding between polymer medium and HNT. The approximations of the advanced model are linked to the numerous experimented data and the stimuli of all factors on the modulus of system are clarified. Halpin-Tsai model overestimates the moduli of system, whereas the advanced model produces the accurate results matching to the experimental facts. “Gc” inversely governs the modulus of system. The variation of “Gc” from 1 to 5 GPa changes the nanocomposite’s modulus by 57%. The interfacial shear modulus (Gi) positively controls the modulus of system and Gi = 20 GPa improves the modulus of samples by 63%.

Effect of temperature anisotropy on residual zonal flow level

Residual level (RZF) of long wavelength zonal flows in high aspect ratio collisionless toroidal plasma with a bi-Maxwellian distribution is calculated using the modern bounce-kinetic theory of neoclassical polarization density [L. Wang and T. S. Hahm, Phys. Plasmas 16, 062309 (2009)]. The result which generalizes the well-known formula for the Maxwellian distribution function [M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998)], RZF=[1+1.63q2ϵ−1/2(T⊥/T∥)3/2]−1, shows a strong dependence on the temperature ratio. Here T⊥ and T∥ are the temperatures in the perpendicular and parallel directions with respect to magnetic field, ϵ is the local inverse aspect ratio, and q is the safety factor. This scaling can be understood from the scalings of barely trapped/passing particles' population fraction and their radial orbit width as well as those of the second adiabatic invariant and the bounce/transit frequency.

Analytic guiding center formulas for bounce-transit motion in a concentric circular, finite inverse aspect ratio tokamak geometry

Bounce-transit motion in concentric circular magnetic geometry is typically analyzed in the limit that the inverse aspect ratio, ε, is small. We prove that this approximation is not necessary to study a concentric circular geometry by deriving new analytical formulas while retaining a non-zero ε. We use these formulas to demonstrate that the approximation is robust for ε≲0.3.

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Shaping effects on scrape-off layer plasma turbulence: A rigorous validation of three-dimensional simulations against TCV measurements

The effect of plasma shaping on scrape-off layer (SOL) plasma turbulence is investigated through a rigorous validation exercise. Two- and three-dimensional simulations of the SOL plasma dynamics in three TCV limited discharges are carried out with the GBS code. These discharges realize an almost circular magnetic equilibrium, an elongated equilibrium, and an elongated equilibrium with negative triangularity. For the three plasma discharges, three simulations are performed, considering (i) a three-dimensional model with an explicit dependence on elongation, triangularity, and the inverse aspect ratio; (ii) a circular three-dimensional model in the infinite aspect ratio limit; and (iii) a two-dimensional model, which assumes a circular magnetic equilibrium in the infinite aspect ratio limit, cold ions, and interchange driven turbulence in the sheath limited regime. Ten validation observables common to simulations and experimental measurements from a reciprocating probe located at the TCV outer midplane are identified, and the agreement between experimental and numerical results relative to each observable is evaluated. The composite metric introduced by Ricci et al. [Phys. Plasmas 18, 032109 (2011)] is then used to assess the overall agreement between simulations and experimental measurements. It is found that the shaping model implemented in GBS improves the description of SOL plasma turbulence, taking into account the impact of elongation and triangularity.

Constructing stellarators with quasisymmetry to high order

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis (Phys Fluids B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landremanet al. (J. Plasma Phys., vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform$\unicode[STIX]{x1D704}>0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are${<}2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

Derivation of bootstrap current fraction scaling formula for 0-D system code analysis

We constructed new scaling formulas for the bootstrap current faction fBS for tokamaks with full non-inductive operation with positive shear safety factor q profiles. A fitting database was built by the ACCOME code, which contains 12,150 cases of self-consistent equilibria with the q profiles satisfying qρ≥1.0 and q95≥3.0 in 0≤ρ<1, where ρ is the normalized minor radius and q95 is q at the 95% poloidal flux surface. From the fitting database, we derived the formulas by the multiple regression analysis. The best formula contained the inverse aspect ratio ε, the poloidal beta, the internal inductance, q95/q0, and the profile shape indices for density and temperature as input. The Root Mean Squared Error for the fitting database of this formula was 0.025. We tested the derived formulas to three kinds of datasets and compared with the K. Gi models and the ITER Physics Design Guidelines model. The first test is for the compact neutron source with the same plasma size with the fitting database, the second (including reversed shear) and the third ones (positive shear only) are for the conventional reactor size. The best formula seems to be applicable to tokamak plasmas with a wide range of parameters.

Fast Recursive Reconnection and the Hall Effect: Hall-MHD Simulations

Abstract Magnetohydrodynamic (MHD) theory and simulations have shown that reconnection is triggered via a fast “ideal” tearing instability in current sheets whose inverse aspect ratio decreases to , with S as the Lundquist number defined by the half-length L of the current sheet (of a thickness of 2a). Ideal tearing, in 2D sheets, triggers a hierarchical collapse via stretching of X-points and recursive instability. At each step, the local Lundquist number decreases, until the subsequent sheet thickness either approaches kinetic scales or the Lundquist number becomes sufficiently small. Here we carry out a series of Hall-MHD simulations to show how the Hall effect modifies recursive reconnection once the ion inertial scale is approached. We show that as the ion inertial length becomes of the order of the inner, singular layer thickness at some step of the recursive collapse, reconnection transits from the plasmoid-dominant regime to an intermediate plasmoid+Hall regime and then to the Hall-dominant regime. The structure around the X-point, the reconnection rate, the dissipation property, and the power spectra are also modified significantly by the Hall effect.

Magnetic flux coordinates for analytic high-beta tokamak equilibria with flow

Magnetic flux coordinates are constructed from an analytic solution (Ito and Nakajima 2009 Plasma Phys. Control. Fusion 51 035007) for the reduced magnetohydrodynamics (MHD) equilibrium equations for high-beta tokamaks in the presence of poloidal and toroidal flows comparable to the poloidal sound velocity. The analytic solution indicates non-circular magnetic flux surfaces and transition between sub- and super-sonic poloidal flows. The magnetic flux coordinates for such non-circular magnetic flux surfaces are obtained for stability analysis. The flux coordinates are numerically obtained from the high order polynomial equations for the relation with the geometrical coordinates. As applications, pressure profiles in the poloidal direction on each flux surface, which become non-constant due to flow, and the flux average of the pressure are obtained. The transitions of the pressure profiles and the flux average of the pressure between sub- and super-sonic poloidal flows are discussed in relation with the radial force balance. The flux coordinates are also obtained analytically by expanding the coordinate relations with respect to the inverse aspect ratio.

Singular global components and frequency shift of the geodesic acoustic continuum modes in shaped tokamaks

Within the ideal magnetohydrodynamic (MHD) model, the geodesic acoustic modes (GAMs) in tokamaks derived by Winsor et al (1968 Phys. Fluids 11 2448) belong to the continuous spectrum, characterised by unbounded non-square integrable eigenfunctions (delta functions) at the singular surfaces (Goedbloed 1975 Phys. Fluids 18 1258). The eigenfunctions of the MHD continua in cylindrical as well as toroidal plasmas include, in addition, components that exist outside the singular surfaces and have singularities of type or where is a flux function that labels the magnetic surfaces, and defines the singular surface (Pao 1975 Nucl. Fusion 15 631). Using a large aspect ratio approximation of tokamak plasmas it is shown in this paper that the GAMs indeed include such singular components. Hence, in addition to the non-square integrable and components of the plasma flow and of the density and pressure perturbations at the GAM surface, the GAM continua also include accompanying and singular components varying as This gives the and components of each GAM in the continuum radially extended profiles and a global character also within ideal theory. To the same order in the expansion, effects of a finite aspect ratio and a non-circular plasma cross section on the GAM frequency are also calculated, and we recover the dependence on inverse aspect ratio and Shafranov shift of the real GAM frequency previously calculated within gyrokinetic theory by Gao (2010 Phys. Plasmas 17 092503). Furthermore, while the dominating shaping effect on the GAM frequency comes from plasma elongation, as shown previously, it is shown in this paper that there is a higher-order triangularity effect that can also be significant. The calculated triangularity effect predicts a nearly linearly increasing GAM frequency with increasing triangularity, a phenomenon observed also in the TCV tokamak.