Toroidal Motion Research Articles

Paleotethys slab pull, self-lubricated weak lithospheric zones, poloidal and toroidal plate motions, and Gondwana tectonics

Abstract The Gondwana megacontinent was composed of different domains separated by self-lubricated weak lithospheric zones, two of which could have extended into Laurasia. Displacement vectors determined through three consecutive paleomagnetism-constrained paleogeographic reconstructions (Early Pennsylvanian–early Guadalupian, ca. 320–270 Ma; late Guadalupian–Middle Triassic, ca. 260–240 Ma; and Late Triassic–early Late Jurassic, ca. 230–160 Ma) show similar orientations to coeval tectonic stresses along Gondwana. Triggered by slab pull at the northern subduction margin of the Paleotethys Ocean, differential displacements between the Gondwana domains caused localized deformation along their borders, reactivating old weak lithospheric zones (e.g., Ventana fold belt south of Buenos Aries province, Argentina; basins such as Cuvette in central Africa; and Neuquén on the Pacific margin of Gondwana). We propose that the wide extent of these structures was possible due to the transmission of mantle toroidal flow induced by strike-slip movements along these focused self-lubricated weak lithospheric zones, along with the northward drift of Pangea. These processes occurred simultaneously with a major mantle reorganization from a huge cold downwelling to a hot upwelling event caused by thermal energy storage beneath Pangea.

Toroidal resonance: Relation to pygmy mode, vortical properties, and anomalous deformation splitting

We review a recent progress in investigation of the isoscalar toroidal dipole resonance (TDR). A possible relation of the TDR and low-energy dipole strength (also called a pygmy resonance) is analyzed. It is shown that the dipole strength in the pygmy region can by understood as a local manifestation of the collective vortical toroidal motion at the nuclear surface. Application of the TDR as a measure of the nuclear dipole vorticity is discussed. Finally, an anomalous splitting of the TDR in deformed nuclei is scrutinized.

Circular orbits and related quasiharmonic oscillatory motion of charged particles around weakly magnetized rotating black holes

We study motion of charged particles in the field of a rotating black hole immersed into an external asymptotically uniform magnetic field, focusing on the epicyclic quasi-circular orbits near the equatorial plane. Separating the circular orbits into four qualitatively different classes according to the sign of the canonical angular momentum of the motion and the orientation of the Lorentz force, we analyse the circular orbits using the so called force formalism. We find the analytical solutions for the radial profiles of velocity, specific angular momentum and specific energy of the circular orbits in dependence on the black hole dimensionless spin and the magnetic field strength. The innermost stable circular orbits are determined for all four classes of the circular orbits. The stable circular orbits with outward oriented Lorentz force can extend to radii lower than the radius of the corresponding photon circular geodesic. We calculate the frequencies of the harmonic oscillatory motion of the charged particles in the radial and vertical directions related to the equatorial circular orbits and study the radial profiles of the radial, $\omega_\mir$, vertical, $\omega_\mit$, and orbital, $\omega_\mip$, frequencies, finding significant differences in comparison to the epicyclic geodesic circular motion. The most important new phenomenon is existence of toroidal charged particle epicyclic motion with $\omega_\mir \sim \omega_{\mit} >> \omega_{\mip}$ that could occur around retrograde circular orbits with outward oriented Lorentz force. We demonstrate that for the rapidly rotating black holes the role of the 'Wald induced charge' can be relevant.

Simulation and Testing of a Modified Mould on UGITECH’s ESRR™

Since the end of 2007, UGITECH S.A. has been equipped with an ESRR™ plant provided by INTECO Special Melting Technologies GmbH. This technology uses a superheated slag to remelt electrodes, in a T-shaped mould equipped with an upper graphite current collector: the current is split into two separate paths. This feature makes the ESRR™ process specifically appropriate to remelt a main electrode into smaller products like billets, with a rather high productivity. Nevertheless, the unique need of UGITECH S.A. to remelt continuous casting blooms, while keeping their 205 × 205 mm section constant, implies quite a small section electrode compared to the normal ESRR™ practice. The initial mould configuration of the machine had to be modified to take better into account this specific demand. The ESRR™ plant is a complex process which combines many simultaneous phenomena—electromagnetic, heat transfer, fluid flow, etc.—with a strong coupling. The first numerical model of this process was built, and data taken from literature [2-15] helped to develop its first heat transfer simulation. Despite the necessary simplifications and work hypotheses, this approach has given these first important results: to confirm the presence of organized toroidal motions in the slag, to provide orders of magnitude for variables one cannot measure in situ (slag flow velocity, temperature distribution), and to compare the ESRR™ mould in its original configuration with a new design. Several industrial trials, performed in the target configuration and simulated with a modified current upper collector, have demonstrated obvious improvements both on the surface quality of the ingots (directly related to the temperature at the metal meniscus) and on the stability of the machine’s regulation (melting speed and electrode immersion). After this confirmation of the advantages of the new configuration, UGITECH S.A. finally placed an order to INTECO GmbH for a new design mould and an associated body.

Simulation Study of Toroidal Flow Generation of Minority Ions by Local ICRF Heating

The toroidal flow generation of minority ions by the local ion cyclotron range of frequencies (ICRF) heating is investigated in a tokamak plasma by applying the GNET code, which can solve the drift kinetic equation in the 5-D phase space. An asymmetry of velocity distribution function in the parallel direction is found and two types of toroidal averaged flow of minority ions are observed. One is the sheared flow near the RF power absorption region depending on the sign of k||, and the other is the toroidal flow, which is larger than the previous one, independent of the sign of k||. It is found that the k||-sign-independent toroidal flow is generated by the net toroidal motion of energetic tail ions and that the k||-sign-dependent flow is related to the mechanism proposed by Ohkawa [Phys. Plasmas 12, 094506 (2005)].

Fluid flow and mixing in linear GTA welding of dissimilar ferrous alloys

In this investigation the heat, fluid flow, and mass transfer for linear autogenous gas tungsten arc (GTA) welding of austenitic stainless steel to low carbon steel is presented. A numerical model was built in COMSOL Multiphysics which included the electromagnetic field, fluid flow, and heat and mass transfer. The results show that there is a toroidal fluid motion at the center of the weld pool, which circulates the molten base metals and completely mixes them with each other. The numerical predictions of the weld-pool shape and the concentrations of iron (Fe), chromium (Cr) and nickel (Ni) were compared to the experimental determinations of the weld-zone and the concentrations of Fe, Cr and Ni. Both numerical and experimental results show that the alloying elements are uniformly mixed throughout the weld zone. The numerical predictions are quantitatively in reasonable agreement with the experimentally determined results.

Alpha particle transport in the presence of ballooning type electrostatic driftwaves

Employing Hamiltonian mechanics, the transport of fusion born alpha particles in the presence of driftwave turbulence is investigated. An analytical turbulence model based on the toroidal drift eigenmode is employed for guiding center orbit-following calculations. It is shown that high energy particles are less susceptible to driftwave turbulence. The passing particle transport is due to overlapping of guiding center electric islands whose widths are inversely proportional to the square root of the parallel velocity. For trapped particles, through a coordinate transformation from the poloidal angle and the parallel velocity to the action-angle variables, the resonance between the bounce motion and the toroidal precession motion, which can cause secondary island formation in the phase space, is demonstrated.

Droplet behavior in a Taylor vortex

Here we present the study on the behavior of individual droplets in a Taylor vortex of immiscible working fluid. The droplets (water or ethanol with the volume of 15–30μL) were introduced into a Taylor–Couette device with an aspect ratio of 6 and a radius ratio of 0.67, in which a mineral oil (density of 0.86g/cm3 and viscosity of 0.066Pas) was used as the working fluid. This configuration ensures a laminar Taylor vortex flow with no occurrence of wavy vortex in the entire operating range. The behavior of the droplets was investigated with a high speed camera and a phase Doppler particle analysis (PDPA) system. The water droplet can be trapped in the vortex center at low Reynolds numbers, corresponding to a circular trajectory, which gradually develops into a three-dimensional toroidal motion with the increasing Reynolds number. Differently, the ethanol droplet exhibits the toroidal motion at low Reynolds numbers and follows the circular trajectory at high Reynolds numbers. Due to their deformability, both water and ethanol droplets are elongated as an ellipsoid shape when moving in the gap, although its length-to-diameter ratio varies along the annulus. Furthermore, the ethanol droplet subject to a sudden-start of the fluid can break into lots of small droplets with the size of microns, which later coalesce into a single droplet again after the Taylor vortex flow becomes stable. The study can help to better understand the droplet behaviors in a Taylor vortex and therefore benefit possible applications conducted in Taylor vortex devices.

Dust devil generation

The equations describing axi-symmetric nonlinear internal gravity waves in an unstable atmosphere are derived. A hydrodynamic model of a dust devil generation mechanism in such an atmosphere is investigated. It is shown that in an unstably stratified atmosphere the convective plumes with poloidal motion can grow exponentially. Furthermore, it is demonstrated that these convective plumes in an atmosphere with weak large scale toroidal motion are unstable with respect to three-dimensional dust devil generation.

Three‐dimensional mantle circulations and lateral slab deformation in the southern Chilean subduction zone

AbstractThe along‐strike variation in plate properties is a common feature in the subduction zones. The southern Chilean subduction is a notable example where the slab age and length, and the thickness of the overriding plate, vary substantially in the trench‐parallel direction. In this study the combined effects of these factors are examined using regional subduction zone models. In the models lateral slab deformation and along‐arc pressure gradient in the mantle occur in response to the differential slab retrograde motion due to variations in slab buoyancy and slab strength. The lower portion of the deep segments acts as a slab edge in part, while rollback of the neighboring, shallower segments influences the toroidal motion induced by the adjacent deep segments. These factors together give rise to complex flow patterns and lead to a great extent of trench‐parallel components in the mantle wedge and subslab mantle and significant upwelling in the back arc. Mantle circulations are characterized by variable length scales for both toroidal and poloidal motions extending over ~ 1500 km. Such three‐dimensional circulations may lead to complicated patterns of seismic anisotropy. The upwelling may result in decompression melting to cause the extensive Patagonian plateau basalts. Our model results indicate that certain segments of the Antarctic slab reach at least 100 km depth. The trajectories of passive tracers show intricate patterns such as helical streamlines and suggest that a recent input from the adjacent subduction zone for slab‐derived components of lavas from the Chilean Ridge cannot be ruled out.

Dense Plasma Focus: A question in search of answers, a technology in search of applications

Diagnostic information accumulated over four decades of research suggests a directionality of toroidal motion for energetic ions responsible for fusion neutron production in the Dense Plasma Focus (DPF) and existence of an axial component of magnetic field even under conditions of azimuthal symmetry. This is at variance with the traditional view of Dense Plasma Focus as a purely irrotational compressive flow. The difficulty in understanding the experimental situation from a theoretical standpoint arises from polarity of the observed solenoidal state: three independent experiments confirm existence of a fixed polarity of the axial magnetic field or related azimuthal current. Since the equations governing plasma dynamics do not have a built-in direction, the fixed polarity must be related with initial conditions: the plasma dynamics must interact with an external physical vector in order to generate a solenoidal state of fixed polarity. Only four such external physical vectors can be identified: the earth's magnetic field, earth's angular momentum, direction of current flow and the direction of the plasma accelerator. How interaction of plasma dynamics with these fields can generate observed solenoidal state is a question still in search of answers; this paper outlines one possible answer. The importance of this question goes beyond scientific curiosity into technological uses of the energetic ions and the high-power-density plasma environment. However, commercial utilization of such technologies faces reliability concerns, which can be met only by first-principles integrated design of globally-optimized industrial-quality DPF hardware. Issues involved in the emergence of the Dense Plasma Focus as a technology platform for commercial applications in the not-too-distant future are discussed.

Toroidal drift precession and wave–particle interaction in shaped tokamaks with finite beta and neoclassical equilibrium effects

The drift kinetic equation is analysed for linear instabilities that are sensitive to the toroidal drift motion of long-mean free path particle populations. The interaction Lagrangian is resolved, and the toroidal drift precession evaluated for realistic tokamak equilibria, including the effects of cross section shaping and finite beta. The drift kinetic equation is expanded around a flux surface in accordance with a neoclassically resolved equilibrium and coincident bootstrap current. Analytical results pertaining to the effect of shaping, magnetic shear and finite beta on the toroidal drift and bounce/transit frequencies of passing and trapped particles are shown.

Generation of plate tectonics with two-phase grain-damage and pinning: Source–sink model and toroidal flow

The grain-damage and pinning mechanism of Bercovici and Ricard (2012) for lithospheric shear-localization is employed in two-dimensional flow calculations to test its ability to generate toroidal (strike-slip) motion and influence plate evolution. This mechanism posits that damage to the interface between phases in a polycrystalline material like peridotite (composed primarily of olivine and pyroxene) increases the number of small Zener pinning surfaces, which then constrain mineral grains to ever smaller sizes, regardless of creep mechanism. This effect allows a self-softening feedback in which damage and grain-reduction can co-exist with a grain-size dependent diffusion creep rheology; moreover, grain growth and weak-zone healing are greatly impeded by Zener pinning thereby leading to long-lived relic weak zones. The fluid dynamical calculations employ source–sink driven flow as a proxy for convective poloidal flow (upwelling/downwelling and divergent/convergent motion), and the coupling of this flow with non-linear rheological mechanisms excites toroidal or strike-slip motion. The numerical experiments show that pure dislocation-creep rheology, and grain-damage without Zener pinning (as occurs in a single-phase assemblages) permit only weak localization and toroidal flow; however, the full grain-damage with pinning readily allows focussed localization and intense, plate-like toroidal motion and strike-slip deformation. Rapid plate motion changes are also tested with abrupt rotations of the source–sink field after a plate-like configuration is developed; the post-rotation flow and material property fields retain memory of the original configuration for extensive periods, leading to suboptimally aligned plate boundaries (e.g., strike-slip margins non-parallel to plate motion), oblique subduction, and highly localized, weak and long lived acute plate-boundary junctions such as at what is observed at the Aleutian–Kurile intersection. The grain-damage and pinning theory therefore readily satisfies key plate-tectonic metrics of localized toroidal motion and plate-boundary inheritance, and thus provides a predictive theory for the generation of plate tectonics on Earth and other planets.

Manifestation of Constrained Dynamics in a Low-Pressure Spark

Some features of neutron emission from dense plasma focus suggest that the participating deuterons have energy in the range of 105 eV and have a directionality of toroidal motion. Theoretical models of these devices assume that the plasma evolves through a purely irrotational flow and thus fail to predict such solenoidal flow on the scale of the plasma dimensions. Predictions of a relaxation theory are consistent with experimental data [S K H Auluck, Physics of Plasmas,18, 032508 (2011)], but the assumptions upon which it is based are not compatible with known features of these devices. There is thus no satisfactory theoretical construct which provides the necessity for solenoidal flow in these devices. This paper proposes such theoretical construct, namely, the principle of constrained dynamics, and describes an experiment which provides support for this idea. The experiment consisted of low inductance, self-breaking spark discharge in helium at a pressure ~100 hPa between two pointed electrodes separated by 30-50 mm distance kept inside a vacuum chamber mounted on a low inductance high voltage capacitor. The current derivative signal showed reproducible sharp dips at all the extrema of the damped sinusoidal discharge. A planar diamagnetic loop centered with and perpendicular to the discharge axis consistently showed a signal representing rate of change of axial magnetic flux. A qualitative explanation of observed phenomena is obtained using a simple model using the proposed principle.

Toroidal runaway beams

We investigate the dynamics of a special group of runaway electrons which possibly play an important role in toroidal fusion devices. Starting from the torus center they are accelerated by a toroidal electric field and are hence forced to move across the toroidal magnetic field into regions with rising poloidal field in order to compensate for the centrifugal forces. Can such particles finally form a tight beam of relativistic runaways in the outboard region or is this prevented due to the perpendicular momentum they gain by passing the toroidal field? Since neither the energy nor the magnetic momentum of the particles is conserved this question has been treated by invoking the relativistic equations of motion. It turns out, however, that the problem can be essentially simplified since, apart from the centrifugal forces associated with the toroidal motion, the inertia forces are negligible. The resulting first order equation can be solved analytically. From the solution it is concluded that the formation of narrow runaway beams with diameters in the range of micrometers and very small pitch angles (v(perpendicular)/v(||)<10(-6)) appears feasible. Such electrons would perform low-frequency oscillations about three to four orders of magnitude lower than the gyrofrequency in the toroidal field. When passing the maximum poloidal magnetic field strength they are suddenly lost from the plasma region.

Discrete element investigation of flow patterns and segregation in a spheronizer

Powder flow in a spheronizer is characterized by a toroidal motion and the flow patterns depend on the disc velocity and the fill level, but also on the particle properties. This work investigates numerically, with a discrete element method (DEM), the impact of these parameters on segregation patterns and the flow dynamics for bidisperse particle size distributions. Characterization of the segregation, by means of a mixing index and the relation with the shear stress in the toroidal domain, is presented. Characteristics, such as mixing curves, concentration profiles and azimuthal velocity correlations, are discussed. A logarithmic expression has been developed to account for the shear stress on the evolution of the azimuthal velocity inside the particulate bed. The combination of the mixing indexes and the concentration profiles is used to quantify the changes observed on the segregation when the fill level and the rotational rate of the spheronizer disk are varied.

3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method

This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been uncoupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal motion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal relaxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use.

Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere

This paper concentrates on the study of the three-dimensional free vibrations in a homogenous isotropic, viscothermoelastic hollow sphere whose surfaces are subjected to stress free, thermally insulated or isothermal boundary conditions. The use of governing partial differential equations is solved into a coupled system of ordinary differential equations. The equation for toroidal motion gets decoupled from rest of the motion and remains unaffected due to thermal variations. Matrix Fr?benious method of extended power series is employed to obtain the solution. The secular equations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived in the com- pact form. The special cases of spheroidal and toroidal modes of vibrations of a hollow sphere have also been deduced and discussed. In order to explore the characteristics of vibrations the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software tools. The computer simulated results have been presented graphically for copper material.

Vibration Analysis of Homogeneous Transradially Isotropic Generalized Thermoelastic Spheres

The exact free vibration analysis of stress free or rigidly fixed, thermally insulated/isothermal, transradially (spherically) isotropic thermoelastic solid sphere has been presented in context of nonclassical thermoelasticity. The transradially isotropic is also frequently referred as spherically isotropic in the literature. The basic governing equations of linear generalized thermoelastic, transradially isotropic, sphere have been uncoupled and simplified with the help of Helmholtz decomposition theorem. The formal solution of the coupled system of partial differential equations has been obtained by employing matrix Fröbenius method of extended series. The secular equations for the existence of possible modes of vibrations in the sphere have been derived by employing boundary conditions. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations have also been deduced and discussed. It is found that the toroidal motion gets decoupled from the spheroidal one and remains independent of thermal variations and thermal relaxation time. In order to illustrate the analytical development, the numerical solution of secular equations for spheroidal motion (S-mode) is carried out with respect of magnesium and solid helium spheres. The lowest frequency and damping factor of vibrational modes have been computed with the help of MATLAB programming and the results are presented graphically. The study may find applications in aerospace, navigation, geophysics tribology, and other industries where spherical structures are in frequent use.

Spectral Decomposition of Orbital Tori

The Kolmogorov―Arnold―Moser theorem states that lightly perturbed integrable Hamiltonian systems maintain their multiply periodic, toroidal motion in the phase space. The assertion that Earth-orbiting satellites under the influence of the geopotential mimic this behavior is the underlying premise of this work. This paper focuses on applying trajectory-following spectral methods on selected orbits to decompose them into multiperiodic Fourier series, effectively compressing ephemerides for long-term use. The proposed approach focuses on fitting local spectral structures, denoted as frequency clusters, within the sampled orbital data to the analytical form of the windowed, truncated, continuous Fourier transform. This approach is significantly more numerically efficient than fitting every coefficient within the N-tuple Fourier series simultaneously. Numerical simulations using integrated data yield root-mean-square error in orbital tori fits at 10 m or less per coordinate axis over a one-year period for most low-inclination, low-eccentricity orbits with altitudes lower than 1900 km.