Non-axisymmetric Field Research Articles

On the effects of magnetic perturbations on fast ion losses studied at TEXTOR

One criterion for the ignition of a fusion plasma is sufficient fast ion confinement. A key aspect in that context is the maintainability of good fast ion confinement in the presence of non-axisymmetric fields, such as those found in stellarators and during the application of resonant magnetic perturbations (RMPs) in tokamaks. This paper focuses on the influence of RMPs on the fast ion losses, studied at the Tokamak experiment for technology-oriented research (TEXTOR), a medium-sized device. TEXTOR is equipped with a flexible perturbation coil system, the dynamic ergodic divertor, allowing for the application of static or rotating perturbation fields. A rotating directional probe is used for the radially resolved fast ion loss measurements. The results achieved are presented and discussed for the poloidal/toroidal perturbation modes m/n = 3/1 and 6/2, with static and rotating fields.

Acoustic radiation force due to nonaxisymmetric sound beams incident on spherical viscoelastic scatterers in tissue

The theory for acoustic radiation force on a viscoelastic sphere of arbitrary size in tissue was extended recently to account for nonaxisymmetric incident fields [Ilinskii et al., POMA 19, 045004 (2013)]. A spherical harmonic expansion was used to describe the incident field. This work was specialized at the spring 2014 ASA meeting to focused axisymmetric sound beams with various focal spot sizes and a scatterer located at the focus. The emphasis of the present contribution is nonaxisymmetric fields, either through moving the scatterer off the axis of an axisymmetric beam or through explicitly defining a nonaxisymmetric beam. This is accomplished via angular spectrum decomposition of the incident field, spherical wave expansions of the resulting plane waves about the center of the scatterer, Wigner D-matrix transformations to express these spherical waves in a coordinate system with the polar axis aligned with the desired radiation force component, and finally integration over solid angle to obtain spherical wave amplitudes as required in the theory. Various scatterer sizes and positions relative to the focus are considered, and the effects of changing properties of both the scatterer and the surrounding tissue are examined. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

Evaluation of the toroidal torque driven by external non-resonant non-axisymmetric magnetic field perturbations in a tokamak

The toroidal torque driven by external non-resonant magnetic perturbations (neoclassical toroidal viscosity) is an important momentum source affecting the toroidal plasma rotation in tokamaks. The well-known force-flux relation directly links this torque to the non-ambipolar neoclassical particle fluxes arising due to the violation of the toroidal symmetry of the magnetic field. Here, a quasilinear approach for the numerical computation of these fluxes is described, which reduces the dimension of a standard neoclassical transport problem by one without model simplifications of the linearized drift kinetic equation. The only limiting condition is that the non-axisymmetric perturbation field is small enough such that the effect of the perturbation field on particle motion within the flux surface is negligible. Therefore, in addition to most of the transport regimes described by the banana (bounce averaged) kinetic equation also such regimes as, e.g., ripple-plateau and resonant diffusion regimes are naturally included in this approach. Based on this approach, a quasilinear version of the code NEO-2 [W. Kernbichler et al., Plasma Fusion Res. 3, S1061 (2008).] has been developed and benchmarked against a few analytical and numerical models. Results from NEO-2 stay in good agreement with results from these models in their pertinent range of validity.

Viscous damping of toroidal angular momentum in tokamaks

The Braginskii viscous stress tensor formalism was generalized to accommodate non-axisymmetric 3D magnetic fields in general toroidal flux surface geometry in order to provide a representation for the viscous damping of toroidal rotation in tokamaks arising from various “neoclassical toroidal viscosity” mechanisms. In the process, it was verified that the parallel viscosity contribution to damping toroidal angular momentum still vanishes even in the presence of toroidal asymmetries, unless there are 3D radial magnetic fields.

Time-domain analytical solutions to pulsed eddy current field excited by a probe coil outside a conducting ferromagnetic pipe

Using the second-order vector potential formalism and block matrix, the non-axisymmetric eddy current field induced by a probe coil positioned perpendicularly outside a conducting ferromagnetic pipe is solved analytically. Then, the time-domain expressions of induced voltage and eddy current density in the pipe are obtained through the Laplace inverse transformation, which is carried out by calculating the residues of poles. Furthermore, the diffusion process of pulsed eddy current in the pipe is examined. Finally, the analytical solutions are verified through the experiment results of two steel pipes with different wall thickness.

A CAD-Based Tool for Calculating Power Deposition on Tokamak Plasma-Facing Components

The SMARDDA software library is used to model plasma interaction with complex engineered surfaces. A simple flux-tube model of power deposition necessitates the following of magnetic fieldlines until they meet geometry taken from a CAD (Computer Aided Design) database. Application is made to 1) models of ITER tokamak limiter geometry and 2) MASTU tokamak divertor designs, illustrating the accuracy and effectiveness of SMARDDA, even in the presence of significant nonaxisymmetric ripple field. SMARDDA's ability to exchange data with CAD databases and its speed of execution also give it the potential for use directly in the design of tokamak plasma facing components.

Thermo-elastic analysis of a non-axisymmetrically heated FGPM hollow cylinder under multi-physical fields

This study deals with the two-dimensional steady-state response of a cylindrical shell made from functionally graded piezoelectric materials under multi-physical fields. The hollow cylinder is subjected to mechanical, electrical and non-axisymmetric thermal fields. Material properties vary continuously through the thick direction of the cylinder based on the power-law distribution of the reinforcement volume fraction. Using equations of equilibrium, strain–displacement and stress–strain relations in conjunction with the potential–displacement equation, a set of coupled differential equations is derived which are then numerically solved by the differential quadrature method to obtain the mechanical stresses in cylinder. In order to obtain the thermal stress distributions in the non-axisymmetrically heated cylinder the complex variable method is applied. The effects of external pressure and material in-homogeneity on the stresses, mechanical and electrical displacements and electric potential distributions are discussed in details. It has been found that increasing the external pressure of the cylinder decreases the electric displacement, justifying industrial application of such material as efficient actuators and sensors. Moreover, dimensionless Von Mises thermal stresses decreases with increasing the material in-homogeneity.

The effect of the plasma position control system on the three-dimensional distortion of the plasma boundary when magnetic perturbations are applied in MAST

When resonant magnetic perturbations are applied in MAST, the plasma edge boundary experiences a three-dimensional (3D) distortion, which can be a few percent of the minor radius in amplitude, in good agreement with ideal 3D equilibrium modelling. This displacement occurs in plasmas both with radial position feedback control applied, and without feedback. When position feedback control is employed, an applied non-axisymmetric field can lead to an exacerbated edge displacement due to an ancillary axisymmetric position correction, with the direction of the correction dependent upon the phase of the applied field with respect to the toroidal position of the sensors used in the controller. This suggests that future machines reliant upon resonant magnetic perturbations for controlling ELMs should consider using a plasma control system capable of applying a position correction which accounts for the non-axisymmetry of applied magnetic perturbations.

Magnetic fields in fully convective M-dwarfs: oscillatory dynamos versus bistability

Abstract M-dwarfs demonstrate two types of activity: (1) strong (kilogauss) almost axisymmetric poloidal magnetic fields; and (2) considerably weaker non-axisymmetric fields, sometimes including a substantial toroidal component. Dynamo bistability has been proposed as an explanation. However, it is not straightforward to obtain such a bistability in dynamo models. On the other hand, the solar magnetic dipole at times of magnetic field inversion becomes transverse to the rotation axis, while the magnetic field becomes weaker at times far from that of inversion. Thus, the Sun resembles a star with the second type of activity. We suggest that M-dwarfs can have magnetic cycles, and that M-dwarfs with the second type of activity can just be stars observed at times of magnetic field inversion. Then the relative number of M-dwarfs with the second type of activity can be used in the framework of this model to determine parameters of stellar convection near the surface.

Measurement of tokamak error fields using plasma response and its applicability to ITER

The nonlinear response of a low-beta tokamak plasma to non-axisymmetric fields offers an alternative to direct measurement of the non-axisymmetric part of the vacuum magnetic fields, often termed ‘error fields’. Possible approaches are discussed for determination of error fields and the required current in non-axisymmetric correction coils, with an emphasis on two relatively new methods: measurement of the torque balance on a saturated magnetic island, and measurement of the braking of plasma rotation in the absence of an island. The former is well suited to ohmically heated discharges, while the latter is more appropriate for discharges with a modest amount of neutral beam heating to drive rotation. Both can potentially provide continuous measurements during a discharge, subject to the limitation of a minimum averaging time. The applicability of these methods to ITER is discussed, and an estimate is made of their uncertainties in light of the specifications of ITER's diagnostic systems. The use of plasma response-based techniques in normal ITER operational scenarios may allow identification of the error field contributions by individual central solenoid coils, but identification of the individual contributions by the outer poloidal field coils or other sources is less likely to be feasible.

Effects of diffraction on acoustic radiation force produced by sound beams incident on spherical viscoelastic scatterers in tissue

The theory for acoustic radiation force on a viscoelastic sphere of arbitrary size in tissue was extended at the spring 2013 ASA meeting to account for nonaxisymmetric fields incident on the scatterer [Ilinskii et al., POMA 19, 045004 (2013)]. The results were presented in a form that permits inclusion of as many spherical harmonics as needed to describe the field structure. At the fall 2013 ASA meeting, it was shown that for spheres having sizes up to about one wavelength, only four or five spherical harmonics are required for convergence of the solution when plane waves are incident on the scatterer. At the present meeting, the model is applied to diffracting sound beams incident on the scatterer. The analysis is based on angular spectrum decomposition of the incident field, expansion of the resulting plane waves in spherical waves, then a Wigner transformation of the latter back into spherical coordinates with polar axis coinciding with the beam axis, and finally integration over solid angle to obtain the spherical wave amplitudes used in the theory. Results are presented for different radiation patterns illustrating dependence of the radiation force both on beamwidth and on wavelength relative to the size of the scatterer.

Hybrid Analysis of Gas Annular Seals With Energy Equation

Annular seals are used in turbomachinery to reduce secondary flow between regions of high and low pressure. In a vibrating rotor system, the nonaxisymmetric pressure field developed in the small clearance between the rotor and the seal generate reactionary forces that can affect the stability of the entire rotor system. Traditionally, two analyses have been used to study the fluid flow in seals, bulk-flow analysis and computational fluid dynamics (CFD). Bulk-flow methods are computational inexpensive, but solve simplified equations that rely on empirically derived friction factor coefficients and are moderately accurate. CFD analyses generally provide more accurate results than bulk-flow codes, but solution time can vary between days and weeks. For gas damper seals, these analyses have been developed with the assumption that the flow can be treated as isothermal. However, some experimental studies have shown that the temperature change across the seal can be as much as 37%. Thus, a comprehensive analysis requires the solution of an energy equation. Recently, a new hybrid method that employs a CFD analysis for the zeroth-order flow and a bulk-flow analysis for the first-order, perturbed flow has been developed. This method has shown to compare well with full CFD analysis and experimental data while being computationally efficient. In this study, the previously developed hybrid method is extended to include the effects of nonisothermal flow. The hybrid method with energy equation is then compared with the isothermal hybrid method and experimental data for several test cases of hole-pattern seals and the importance of the use of energy equation is studied.

TRENDS IN ULTRACOOL DWARF MAGNETISM. I. X-RAY SUPPRESSION AND RADIO ENHANCEMENT

Although ultracool dwarfs (UCDs) are now known to generate and dissipate strong magnetic fields, a clear understanding of the underlying dynamo is still lacking. We have performed X-ray and radio observations of seven UCDs in a narrow range of spectral type (M6.5-M9.5) but spanning a wide range of projected rotational velocities (v sin i ~ 3-40 km/s). We have also analyzed unpublished archival Chandra observations of four additional objects. All of the newly-observed targets are detected in the X-ray, while only one is detected in the radio, with the remainder having sensitive upper limits. We present a database of UCDs with both radio and X-ray measurements and consider the data in light of the so-called G\"udel-Benz relation (GBR) between magnetic activity in these bands. Some UCDs have very bright radio emission and faint X-ray emission compared to what would be expected for rapid rotators, while others show opposite behavior. We show that UCDs would still be radio-over-luminous relative to the GBR even if their X-ray emission were at standard rapid-rotator "saturation" levels. Recent results from Zeeman-Doppler imaging and geodynamo simulations suggest that rapidly-rotating UCDs may harbor a bistable dynamo that supports either a stronger, axisymmetric magnetic field or a weaker, non-axisymmetric field. We suggest that the data can be explained in a scenario in which strong-field objects obey the GBR while weak-field objects are radio-over-luminous and X-ray-under-luminous, possibly because of a population of gyrosynchrotron-emitting coronal electrons that is continuously replenished by low-energy reconnection events.

Non-Axisymmetric Flow Fields in Axisymmetric CVD Reactor Setups Revisited: Influence on the Film's Non-Uniformity

Non-Axisymmetric Flow Fields in Axisymmetric CVD Reactor Setups Revisited: Influence on the Film's Non-Uniformity

Analytical solutions to pulsed eddy current field excited by a differently oriented probe coil outside a conducting ferromagnetic pipe

Using the second-order vector potential and Lorentz reciprocity theorem, the non-axisymmetric eddy current field induced by a probe coil outside a conducting ferromagnetic pipe is obtained analytically, where the probe coil is differently oriented: its axis being along the pipe radial direction, parallel to the pipe axis, along the pipe circumferential direction. Then, the time-domain expressions of induced voltage and eddy current density in the pipe are obtained through the Laplace inverse transformation, by calculating the residues of poles. Furthermore, the diffusion process of pulsed eddy current in the pipe, and the detection sensitivity of the time-domain induced voltage signal to the wall thickness are studied. Finally, the analytical solutions are verified through the experimental results of a steel pipe. It is found that when the probe coil is positioned such that its axis is perpendicular to the pipe axis, the strongest and the most sensitive induction voltage signal for detecting the wall thickness of a ferromagnetic pipe is obtained.

M-dwarf stellar winds: the effects of realistic magnetic geometry on rotational evolution and planets

We perform three-dimensional numerical simulations of stellar winds of early-M dwarf stars. Our simulations incorporate observationally reconstructed large-scale surface magnetic maps, suggesting that the complexity of the magnetic field can play an important role in the angular momentum evolution of the star, possibly explaining the large distribution of periods in field dM stars, as reported in recent works. In spite of the diversity of the magnetic field topologies among the stars in our sample, we find that stellar wind flowing near the (rotational) equatorial plane carries most of the stellar angular momentum, but there is no preferred colatitude contributing to mass loss, as the mass flux is maximum at different colatitudes for different stars. We find that more non-axisymmetric magnetic fields result in more asymmetric mass fluxes and wind total pressures $p_{\rm tot}$ (defined as the sum of thermal, magnetic and ram pressures). Because planetary magnetospheric sizes are set by pressure equilibrium between the planet's magnetic field and $p_{\rm tot}$, variations of up to a factor of $3$ in $p_{\rm tot}$ (as found in the case of a planet orbiting at several stellar radii away from the star) lead to variations in magnetospheric radii of about 20 percent along the planetary orbital path. In analogy to the flux of cosmic rays that impact the Earth, which is inversely modulated with the non-axisymmetric component of the total open solar magnetic flux, we conclude that planets orbiting M dwarf stars like DT~Vir, DS~Leo and GJ~182, which have significant non-axisymmetric field components, should be the more efficiently shielded from galactic cosmic rays, even if the planets lack a protective thick atmosphere/large magnetosphere of their own.

Role of plasma response in displacements of the tokamak edge due to applied non-axisymmetric fields

Linear, two-fluid, resistive modelling of the plasma response to applied non-axisymmetric fields shows significant displacement of edge temperature and density profiles. The calculated displacements, often of 2 cm or more in H-mode pedestals with parameters appropriate to DIII-D, are due to the helical distortions resulting from stable edge modes being driven to finite amplitude by the applied fields. In many cases, these displacements are greater in magnitude, and different in phase, than the distortions of the separatrix manifolds predicted from vacuum modelling. Comparison of these results with experimental measurements from Thomson scattering and soft x-ray imaging finds good quantitative agreement. In these experiments, the phase of the applied non-axisymmetric magnetic field was flipped or rotated in order to probe the non-axisymmetric features of the response. The poloidal structures measured by x-ray imaging show clear indications of a helical response, as opposed to simply a change in the axisymmetric transport. Inclusion of two-fluid effects and rotation are found to be important in obtaining quantitative agreement with Thomson scattering data. Modelling shows screening of islands in the H-mode pedestal, but island penetration near the top of the pedestal where the electron rotation vanishes in plasmas with co-current rotation. Enhanced transport due to these islands may provide a mechanism for maintaining the pedestal width below the stability threshold of edge-localized modes. For typical DIII-D parameters, it is shown that the linear approximation is often near or beyond the limit of validity in the H-mode edge; however, the general agreement with experimental measurements indicates that these linear results nevertheless maintain good predictive value for profile displacements.

A method for the estimate of the wall diffusion for non-axisymmetric fields using rotating external fields

A new method for the estimate of the wall diffusion time of non-axisymmetric fields is developed. The method based on rotating external fields and on the measurement of the wall frequency response is developed and tested in EXTRAP T2R. The method allows the experimental estimate of the wall diffusion time for each Fourier harmonic and the estimate of the wall diffusion toroidal asymmetries. The method intrinsically considers the effects of three-dimensional structures and of the shell gaps. Far from the gaps, experimental results are in good agreement with the diffusion time estimated with a simple cylindrical model that assumes a homogeneous wall. The method is also applied with non-standard configurations of the coil array, in order to mimic tokamak-relevant settings with a partial wall coverage and active coils of large toroidal extent. The comparison with the full coverage results shows good agreement if the effects of the relevant sidebands are considered.

Comparisons of linear and nonlinear plasma response models for non-axisymmetric perturbations

With the installation of non-axisymmetric coil systems on major tokamaks for the purpose of studying the prospects of ELM-free operation, understanding the plasma response to the applied fields is a crucial issue. Application of different response models, using standard tools, to DIII-D discharges with applied non-axisymmetric fields from internal coils, is shown to yield qualitatively different results. The plasma response can be treated as an initial value problem, following the system dynamically from an initial unperturbed state, or from a nearby perturbed equilibrium approach, and using both linear and nonlinear models [A. D. Turnbull, Nucl. Fusion 52, 054016 (2012)]. Criteria are discussed under which each of the approaches can yield a valid response. In the DIII-D cases studied, these criteria show a breakdown in the linear theory despite the small 10−3 relative magnitude of the applied magnetic field perturbations in this case. For nonlinear dynamical evolution simulations to reach a saturated nonlinear steady state, appropriate damping mechanisms need to be provided for each normal mode comprising the response. Other issues arise in the technical construction of perturbed flux surfaces from a displacement and from the presence of near nullspace normal modes. For the nearby equilibrium approach, in the absence of a full 3D equilibrium reconstruction with a controlled comparison, constraints relating the 2D system profiles to the final profiles in the 3D system also need to be imposed to assure accessibility. The magnetic helicity profile has been proposed as an appropriate input to a 3D equilibrium calculation and tests of this show the anticipated qualitative behavior.

Acoustic radiation force on a sphere without restriction to axisymmetric fields

The analysis presented at the previous ASA meeting related to investigation of the acoustic radiation force on a sphere embedded in a soft elastic medium with shear modulus that is several orders of magnitude smaller than its bulk modulus. The acoustic field was assumed to be axisymmetric and the spherical scatterer to be located on the axis of the acoustic beam. When one of these conditions is violated, the problem loses its symmetry. In this talk, the acoustic radiation force is considered in the more general case of nonaxisymmetric fields. The calculation is performed in Lagrangian coordinates. All acoustic fields, incident as well as scattered, depend on all three spherical coordinates. The incident and scattered waves, which include both potential and solenoidal parts, are expanded with respect to spherical harmonics. An analytical expression for the acoustic radiation force derived in this investigation may contain as many spherical harmonics as needed. In limiting cases when the scatterer is in liquid and only two modes, monopole and dipole, remain in the scattered fields, the solution for the acoustic radiation force recovers the results reported by Gor’kov [Sov. Phys. Doklady 6, 773 (1962)]. [Work supported by NIH DK070618 and EB011603.]