Axisymmetric Torus Research Articles

Ripple transport and neoclassical diffusion in IR-T1 tokamak

In tokamaks, small variations in the magnetic field create ripple. The discontinuous nature of the magnetic field coils in an axisymmetric torus conduces to additional particle trapping, and it is responsible for an additional neoclassical diffusion. Ripples also reduce the particle removal efficiency and disturb plasma confinement and cause constraints in the design of magnet of fusion reactor. Therefore, it is quite important to include the ripple for the design of plasma edge components. Herein, several considerations are taken into account to calculate and evaluate the diffusion coefficient and ion heat conductivity in ripple transport and also to compare it with neoclassical mode.

Neoclassical, semi-collisional tearing mode theory in an axisymmetric torus

A set of layer equations for determining the stability of semi-collisional tearing modes in an axisymmetric torus, incorporating neoclassical physics, in the small ion Larmor radius limit, is provided. These can be used as an inner layer module for inclusion in numerical codes that asymptotically match the layer to toroidal calculations of the tearing mode stability index, $\unicode[STIX]{x1D6E5}^{\prime }$. They are more complete than in earlier work and comprise equations for the perturbed electron density and temperature, the ion temperature, Ampère’s law and the vorticity equation, amounting to a twelvth-order set of radial differential equations. While the toroidal geometry is kept quite general when treating the classical and Pfirsch–Schlüter transport, parallel bootstrap current and semi-collisional physics, it is assumed that the fraction of trapped particles is small for the banana regime contribution. This is to justify the use of a model collision term when acting on the localised (in velocity space) solutions that remain after the Spitzer solutions have been exploited to account for the bulk of the passing distributions. In this respect, unlike standard neoclassical transport theory, the calculation involves the second Spitzer solution connected with a parallel temperature gradient, because this stability problem involves parallel temperature gradients that cannot occur in equilibrium toroidal transport theory. Furthermore, a calculation of the linearised neoclassical radial transport of toroidal momentum for general geometry is required to complete the vorticity equation. The solutions of the resulting set of equations do not match properly to the ideal magnetohydrodynamic (MHD) equations at large distances from the layer, and a further, intermediate layer involving ion corrections to the electrical conductivity and ion parallel thermal transport is invoked to achieve this matching and allow one to correctly calculate the layer $\unicode[STIX]{x1D6E5}^{\prime }$.

Double Arc Instability in the Solar Corona

Abstract The stability of the magnetic field in the solar corona is important for understanding the causes of solar eruptions. Although various scenarios have been suggested to date, the tether-cutting reconnection scenario proposed by Moore et al. is one of the widely accepted models to explain the onset process of solar eruptions. Although the tether-cutting reconnection scenario proposes that the sigmoidal field formed by internal reconnection is the magnetic field in the pre-eruptive state, the stability of the sigmoidal field has not yet been investigated quantitatively. In this paper, in order to elucidate the stability problem of the pre-eruptive state, we developed a simple numerical analysis in which the sigmoidal field is modeled by a double arc electric current loop and its stability is analyzed. As a result, we found that the double arc loop is more easily destabilized than the axisymmetric torus, and it becomes unstable even if the external field does not decay with altitude, which is in contrast to the axisymmetric torus instability. This suggests that tether-cutting reconnection may well work as the onset mechanism of solar eruptions, and if so, the critical condition for eruption under a certain geometry may be determined by a new type of instability rather than by the torus instability. Based on them, we propose a new type of instability called double arc instability (DAI). We discuss the critical conditions for DAI and derive a new parameter κ, defined as the product of the magnetic twist and the normalized flux of the tether-cutting reconnection.

Subcritical versus supercritical transition to turbulence in curved pipes

Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen–Poiseuille flow if both the amplitude of flow perturbations and the Reynolds number $\mathit{Re}$ exceed a minimum threshold (subcritical transition). As the pipe curvature increases, centrifugal effects become important, modifying the basic flow as well as the most unstable linear modes. If the curvature (tube-to-coiling diameter $d/D$) is sufficiently large, a Hopf bifurcation (supercritical instability) is encountered before turbulence can be excited (subcritical instability). We trace the instability thresholds in the $\mathit{Re}-d/D$ parameter space in the range $0.01\leqslant d/D\leqslant 0.1$ by means of laser-Doppler velocimetry and determine the point where the subcritical and supercritical instabilities meet. Two different experimental set-ups are used: a closed system where the pipe forms an axisymmetric torus and an open system employing a helical pipe. Implications for the measurement of friction factors in curved pipes are discussed.

Flux tube train model for local turbulence simulation of toroidal plasmas

A new simulation method for local turbulence in toroidal plasmas is developed by extending the conventional idea of the flux tube model. In the new approach, a train of flux tubes is employed, where flux tube simulation boxes are serially connected at each end along a field line so as to preserve a symmetry of the local gyrokinetic equations for image modes in an axisymmetric torus. Validity of the flux tube train model is confirmed against the toroidal ion temperature gradient turbulence for a case with a long parallel correlation of fluctuations, demonstrating numerical advantages over the conventional method in the time step size and the symmetry-preserving property.

Magnetic surfaces in an axisymmetric torus

A method is developed for specifying the boundary equilibrium magnetic surface in an axially symmetric torus by using the absolute values of the magnetic field B = Bs(θ) and the gradient of the poloidal flux ||∇Ψ| = |∇Ψ|s(θ) in a special flux coordinate system. By setting two surface constants (e.g., the safety factor q and dp/dΨ) and matching the absolute values of the magnetic field and the flux gradient on a closed magnetic surface, it is possible to find all equilibrium magnetic functions (including n · ∇ ln B and the local shear s) and all constants (including the toroidal current J and the shear dμ/dΨ) on this surface. Such a non-traditional formulation of the boundary conditions in solving the stability problem in an axisymmetric torus allows one to impose intentional conditions on plasma confinement and MHD stability at the periphery of the system.

On stability of MHD flows located on the surface of axisymmetric torus

We consider the problem of the evolution of azimuthal perturbations (spontaneous swirling) in axisymmetric magnetohydrodynamic flows of an ideally conducting fluid with circular streamlines. The fluid is in a toroidal gap between two surfaces with constant values of the stream function. The equations of fluid motion are derived in the approximation of narrow gap. The parameters at which spontaneous swirling is possible are determined numerically, and the properties of secondary swirling flows resulting from instability of the initial steady-state poloidal flow are established. It is shown that for certain parameters of the initial poloidal flow, the energy of the initial flow is almost completely converted to the energy of the azimuthal (rotational) velocity field and magnetic field that arise. In this case, over a wide range of parameters of the initial flow, the time-averaged energies of the rotational motion and magnetic field take identical values for large t. We note that even for a small degree of magnetization, instability can give rise to a considerable magnetic field, which can be treated as the spontaneous occurrence of a magnetic field due to extension of the force lines of the initial weak poloidal magnetic field. The numerical studies of spontaneous swirling revealed some properties of the secondary flow, which is irregular (chaotic) in time and its spatial structure is rather complex with the presence of differential rotation, which is a consequence of the conservation laws for the angular momentum and azimuthal magnetic flux.

Numerical computation of the helical Chandrasekhar–Kendall modes

A new formulation is presented for numerically computing the helical Chandrasekhar–Kendall modes in an axisymmetric torus. It explicitly imposes ∇·B=0 and yields a standard matrix eigenvalue problem, which can then be solved by standard matrix eigenvalue techniques. Numerical implementation and computational results are shown for an axisymmetric torus typical of reversed field pinch and spherical tokamak.

Variational coordinate transformation in plasma physics

It is well-known from scaling arguments that action-based field theories do not possess localized solutions in spaces of more than one dimension. The same scaling argument, modified to account for external forces, is applied to magnetic plasma confinement in an axisymmetric torus. It yields an integral solvability condition of some interest.

Observing the Unobservable? Modeling Coronal Cavity Densities

Prominence cavities in coronal helmet streamers are readily detectable in white-light coronagraph images, yet their interpretation may be complicated by projection effects. In order to determine a cavity's density structure, it is essential to quantify the contribution of noncavity features along the line of sight. We model the coronal as an axisymmetric torus that encircles the Sun at constant latitude and fit it to observations of a white-light observed by the Mauna Loa Solar Observatory (MLSO) MK4 coronagraph from 2006 January 25 to 30. We demonstrate that spurious noncavity contributions (including departures from axisymmetry) are minimal enough to be incorporated in a density analysis as conservatively estimated uncertainties in the data. We calculate a radial density profile for material and for the surrounding helmet streamer (which we refer to as the cavity rim) and find that the density is depleted by a maximum of 40% compared to the surrounding helmet streamer at low altitudes (1.18 R☉) but is consistently higher (double or more) than in coronal holes. We also find that the relative density depletion between and surrounding helmet decreases as a function of height. We show that both increased temperature in the relative to the surrounding helmet streamer and a magnetic flux rope configuration might lead to such a flattened density profile. Finally, our model provides general observational guidelines that can be used to determine when a is sufficiently unobstructed to be a good candidate for plasma diagnostics.

Magnetohydrodynamic model of equatorial plasma torus in planetary nebulae

Some basic structures in planetary nebulae are modeled as self-organized magnetohydrodynamic (MHD) plasma configurations with radial flow. These configurations are described by time self-similar dynamics, where space and time dependences of each physical variable are in separable form. Axisymmetric toroidal MHD plasma configuration is solved under the gravitational field of a central star of mass $M$. With an azimuthal magnetic field, this self-similar MHD model provides an equatorial structure in the form of an axisymmetric torus with nested and closed toroidal magnetic field lines. In the absence of an azimuthal magnetic field, this formulation models the basic features of bipolar planetary nebulae. The evolution function, which accounts for the time evolution of the system, has a bounded and an unbounded evolution track governed respectively by a negative and positive energy density constant $H$.

Canonical Hamiltonian theory of the guiding-center motion in an axisymmetric torus, with the different time scales well separated

A new set of exact canonical variables of guiding-center motion in an axisymmetric torus has been systematically identified and an action-angle variable formalism has been established. In this new canonical Hamiltonian theory, the time scale of poloidal motion and the time scale of toroidal drift (precession) of the guiding-centers are well separated.

Chandrasekhar-Kendall modes and Taylor relaxation in an axisymmetric torus

The helicity-conserving Taylor relaxation of a plasma in a toroidal chamber to a force-free configuration, which means j=(j‖∕B)B with j‖∕B independent of position, can be generalized to include the external injection of magnetic helicity. When this is done, j‖∕B has resonant values, which can be understood using the eigenmodes of Taylor-relaxed plasmas enclosed by a perfectly conducting toroidal shell. These eigenmodes include a toroidal generalization of those found by Chandrasekhar and Kendall (CK) [Astrophys. J. 126, 457 (1957)] for a spherical chamber, which has no externally produced magnetic flux. It is shown that the CK modes in an axisymmetric torus are of three types: (1) helical modes as well as axisymmetric modes that have (2) and have no (3) net toroidal flux. Yoshida and Giga (YG) [Math. Z. 204, 235 (1990)] published a fourth class of modes: axisymmetric modes that have no net toroidal flux in the chamber due to toroidal flux produced by a net poloidal current in the shell canceling the net toroidal flux from the plasma currents. Jensen and Chu [Phys. Fluids 27, 2881 (1984)], as well as Taylor [Rev. Mod. Phys. 58, 741 (1986)], considered modes in which the vector potential was zero on the axisymmetric toroidal chamber. It is shown that these Jensen-Chu-Taylor modes include only the CK helical modes and the CK axisymmetric modes without net toroidal flux. If the toroidal chamber is perfectly conducting except for a cut that prevents a net poloidal current from flowing, resonances in j‖∕B occur at the eigenvalues of the axisymmetric CK modes. Jensen and Chu studied this type of resonance. Without the cut, so a poloidal current flows to conserve the net toroidal flux, it is shown that j‖∕B resonances occur at the eigenvalues of the CK modes that have no net toroidal flux and at the eigenvalues of the YG modes, which are upshifted from the eigenvalues of the axisymmetric CK modes that carry net toroidal flux.

Density limit for electron plasmas confined by magnetic surfaces

The density limit for electron plasmas confined by toroidal magnetic surfaces is investigated. In a cylinder, the well-known limit is the Brillouin density, nB≡ϵ0B2∕2me. In an axisymmetric torus, the confining region shifts outward in major radius, and this shift is shown to equal half the plasma radius when n∕nB≈ι2a∕R0, where ι=1∕q is the rotational transform of the magnetic field and a∕R0 is the inverse aspect ratio of the torus. In a nonaxisymmetric torus, electron confinement is found to be lost due to stochasticity effects when n∕nB≈(ι2∕8M2)(a∕R0)2∕δMN. The asymmetry amplitudes δMN are the fractional variations in n∕B2 on a magnetic surface in the poloidal mode number M and the toroidal mode number N≈ιM.

Consideration of magnetic field fluctuation measurements in torus plasma with a heavy ion beam probe

The article proposes a method of magnetic fluctuation measurement using a heavy ion beam probe (HIBP) in an axisymmetric torus configuration. The method is based on the detection of the toroidal position (not velocity) of the secondary beam in the analyzer. However, the method needs careful consideration with respect to path integral fluctuations along the probing beam orbit to evaluate local magnetic fluctuation, similarly to density fluctuation measurements with a HIBP. Here, we present an analytic formula to estimate and calculate the path integral effects for different fluctuation patterns in the profile, the correlation length, the radial wavelength, and the poloidal mode number. As a result, it is found that a large distance between the plasma and the detector lessens the importance of the path integral effect, and that local fluctuation of the magnetic field can be properly detected with a HIBP.

A Classical Model for the Extended Electron I

In order to avoid divergence of its self-energy the free electron at rest is described as a relativistic continuum of finite extent. It has the form of an axisymmetric torus of finite aspect ratio, which rigidly rotates around its axis of symmetry with superluminal speed (v > c). It is shown that there is a class of stationary solutions of the free-boundary value problem. The parameter dependence of these solutions is related to experimental data.

Transport theory for potato orbits in an axisymmetric torus with finite toroidal flow speed

Transport theory for potato orbits in the region near the magnetic axis in an axisymmetric torus such as tokamaks and spherical tori is extended to the situation where the toroidal flow speed is of the order of the sonic speed as observed in National Spherical Torus Experiment [E. J. Synakowski, M. G. Bell, R. E. Bell et al., Nucl. Fusion 43, 1653 (2003)]. It is found that transport fluxes such as ion radial heat flux, and bootstrap current density are modified by a factor of the order of the square of the toroidal Mach number. The consequences of the orbit squeezing are also presented. The theory is developed for parabolic (in radius r) plasma profiles. A method to apply the results of the theory for the transport modeling is discussed.

Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities

We introduce a class of robust soliton clusters composed of N fundamental solitons in three-dimensional media combining the self-focusing cubic and self-defocusing quintic nonlinearities. The angular momentum is lent to the initial cluster through staircase or continuous ramp-like phase distribution. Formation of these clusters is predicted analytically, by calculating an effective interaction Hamiltonian Hint. If a minimum of Hint is found, direct three-dimensional simulations demonstrate that, when the initial pattern is close to the predicted equilibrium size, a very robust rotating cluster does indeed exist, featuring persistent oscillations around the equilibrium configuration (clusters composed of N = 4,5, and 6 fundamental solitons are investigated in detail). If a strong random noise is added to the initial configuration, the cluster eventually develops instability, either splitting into several fundamental solitons or fusing into a nearly axisymmetric vortex torus. These outcomes match the stability or instability of the three-dimensional vortex solitons with the same energy and spin; in particular, the number of the fragments in the case of the break-up is different from the number of solitons in the original cluster, being instead determined by the dominant mode of the azimuthal instability of the corresponding vortex soliton. The initial form of the phase distribution is important too: under the action of the noise, the cluster with the built-in staircase-like phase profile features azimuthal instability, while the one with the continuous distribution fuses into a vortex torus.

On the Structure of the Electron

Usually the electron is described as a mathematical point with charge, mass, spin angular momentum, and electromagnetic field. Because of the unbounded energies this causes mathematical difficulties [1]. These can be avoided by considering a finite radius. For a “free electron at rest” a classical relativistic model is presented where an axisymmetric torus models the electron. This configuration “differentially rotates” around its axis of symmetry with superluminal speed.

Statistical properties of the radial transport in the magnetic field with radially bounded stochastic region

Statistical properties of the radial heat transport of electrons are clarified in the various types of the radially bounded irregular magnetic field inside an axisymmetric torus plasma, where the collisional stochasticity due to the Coulomb collision and the magnetic stochasticity due to a radially bounded perturbed field coexist. Extensive Monte Carlo numerical analysis are performed in the two dimensional parameter space ( s b / s bc , ν/ ν t ) with a delta function initial distribution in the radial direction. The parameters s b and ν are the strength of a magnetic field perturbation and the collision frequency, respectively. The normalization parameter s bc corresponds to the islands overlapping criterion, and ν t is the characteristic frequency of the passing particle orbits in the corresponding regular magnetic field. It is shown that the type of the transport process is closely connected with the locality of the particle radial displacements. As s b / s bc (⩾1) increases and ν/ ν t decreases, the particle distribution spreads out in a whole stochastic region after the short time ballistic (dynamical) phase reflecting the non-locality of radial displacements. As a consequence the subdiffusive process ( s b / s bc ⩾1, ν/ ν t ⩾0) or the uniform mixing process ( s b / s bc ⪢1, ν/ ν t =0) are observed in the long time limit. The standard diffusion approach is applicable only in the limited parameter region when the fairly frequent collisions inside the radially bounded stochastic region with the mixture of regular and irregular domains can recover the locality of particle displacements.