Toroidal Flux Research Articles

Magnetic Fields in Massive Stars. II. The Buoyant Rise of Magnetic Flux Tubes through the Radiative Interior

We present results from an investigation of the dynamical behavior of buoyant magnetic flux rings in the radiative interior of a uniformly rotating, early-type star. Our physical model describes a thin, axisymmetric, toroidal flux tube that is released from the outer boundary of the convective core and is acted on by buoyant, centrifugal, Coriolis, magnetic tension, and aerodynamic drag forces. We find that rings emitted in the equatorial plane can attain a stationary equilibrium state that is stable with respect to small displacements in radius, but is unstable when perturbed in the meridional direction. Rings emitted at other latitudes travel toward the surface along trajectories that largely parallel the rotation axis of the star. Over much of the ascent, the instantaneous rise speed is determined by the rate of heating by the absorption of radiation that diffuses into the tube from the external medium. Since the timescale for this heating varies like the square of the tube cross-sectional radius, for the same field strength, thin rings rise more rapidly than do thick rings. For a reasonable range of assumed ring sizes and field strengths, our results suggest that buoyancy is a viable mechanism for bringing magnetic flux from the core to the surface, being capable of accomplishing this transport in a time that is generally much less than the stellar main-sequence lifetime.

Why helicity injection causes coronal flux tubes to develop an axially invariant cross-section

It is shown that electric current flowing along an axially non-uniform magnetic flux tube produces an associated non-linear, non-conservative axial MHD force which pumps plasma from regions where the flux tube diameter is small to regions where it is large. In particular, this force will ingest plasma into the ends of a fat, initially potential flux tube and then pump the ingested plasma towards the middle bulge, thereby causing mass accumulation at the bulge.The ingested plasma convects frozen-in toroidal magnetic flux which accumulates at the middle as well. Flux accumulation at the bulge has the remarkable consequence of causing the bulge to diminish so that the flux tube becomes axially uniform as observed in coronal loops. Stagnation of the convergent plasma flow at the middle heats the plasma. A small number of tail particles bouncing synchronously between approaching fluid elements can be Fermi-accelerated to very high energies. Since driving a current along a flux tube is tantamount to helicity injection into the flux tube, this mass ingestion, heating, and straightening should be ubiquitous to helicity injection processes.

On the Ejection Mechanism of Bullets in SS 433

We discuss plausible mechanisms to produce bulletlike ejecta from the precessing disk in the SS 433 system. We show that nonsteady shocks in the sub-Keplerian accretion flow can provide the basic timescale of the ejection interval while the magnetic rubber-band effect of the toroidal flux tubes in this disk can yield flaring events.

The formation of sunspots and starspots

The spatial scale of large solar active regions and many of their observed features indicate that they originate from the emergence of a coherent magnetic structure of rather well-ordered toroidal magnetic flux in the solar interior. Numerical simulations of the instability of magnetic flux tubes and their rise through the convection zone are consistent with the basic observed properties of sunspot groups like low emergence latitudes, tilt angles with respect to the east-west direction, and proper motions. The success of this approach in the case of the Sun motivates its application to other magnetically active stars. The effect of the Coriolis force on rising magnetic flux tubes provides an explanation for the existence of high-latitude spots on rapidly rotating active stars. Detailed models have been developed for the distribution of flux emergence latitudes depending on the rotation rate and internal structure of cool stars in various evolutionary stages. The models for very young (T-Tauri-like) stars exhibit extended latitude ranges of flux emergence, including the poles. In the case of rapidly rotating main-sequence stars, the flux tubes appear in mid to high latitudes; neither equatorial nor truly polar flux emergence is found. The trapping of flux tubes in giants with a sufficiently small (relative) core size suggests an explanation for the strong decline of X-ray emission of cool giants across the ‘coronal dividing line’ in the Hertzsprung-Russell diagram. We give an overview of the solar and stellar results and discuss the benefits and the limitations of applying the solar paradigm to active stars.

Poynting Jets from Accretion Disks

We give further considerations on the problem of the evolution of a coronal, force-free magnetic field which threads a differentially rotating, conducting Keplerian disk, extending the work of Li {\it et al.} (2001). This situation is described by the force-free Grad-Shafranov (GS) equation for the flux function $\Psi(r,z)$ which labels the poloidal field lines (in cylindrical coordinates). The GS equation involves a function $H(\Psi)$ describing the distribution of poloidal current which is determined by the differential rotation or {\it twist} of the disk which increases linearly with time. We numerically solve the GS equation in a sequence of volumes of increasing size corresponding to the expansion of the outer perfectly conducting boundaries at ($R_{m}, Z_{m}$). The outer boundaries model the influence of an external non-magnetized plasma. The sequence of GS solutions provides a model for the dynamical evolution of the magnetic field in response to (1) the increasing twist of the disk and (2) the pressure of external plasma. We find solutions with {\it magnetically collimated} Poynting jets where there is a {\it continuous} outflow of energy, angular momentum, and toroidal magnetic flux from the disk into the external space.

Three‐dimensional Magnetohydrodynamic Simulations of an Accretion Disk with Star‐Disk Boundary Layer

We present global 3D MHD simulations of geometrically thin but unstratified accretion disks in which a near Keplerian disk rotates between two bounding regions with initial rotation profiles that are stable to the MRI. The inner region models the boundary layer between the disk and an assumed more slowly rotating central, non magnetic star. We investigate the dynamical evolution of this system in response to initial vertical and toroidal fields imposed in a variety of domains contained within the near Keplerian disk. Cases with both non zero and zero net magnetic flux are considered and sustained dynamo activity found in runs for up to fifty orbital periods at the outer boundary of the near Keplerian disk. Simulations starting from fields with small radial scale and with zero net flux lead to the lowest levels of turbulence and smoothest variation of disk mean state variables. For our computational set up, average values of the Shakura & Sunyaev (1973) $\alpha$ parameter in the Keplerian disk are typically $0.004\pm 0.002.$ Magnetic field eventually always diffuses into the boundary layer resulting in the build up of toroidal field inward angular momentum transport and the accretion of disk material. The mean radial velocity, while exhibiting large temporal fluctuations is always subsonic. Simulations starting with net toroidal flux may yield an average $\alpha \sim 0.04.$ While being characterized by one order of magnitude larger average $\alpha$, simulations starting from vertical fields with large radial scale and net flux may lead to the formation of persistent non-homogeneous, non-axisymmetric magnetically dominated regions of very low density.

Anderson localization of ballooning modes, quantum chaos and the stability of compact quasiaxially symmetric stellarators

The radially local magnetohydrodynamic (MHD) ballooning stability of a compact, quasiaxially symmetric stellarator (QAS), is examined just above the ballooning beta limit with a method that can lead to estimates of global stability. Here MHD stability is analyzed through the calculation and examination of the ballooning mode eigenvalue isosurfaces in the 3-space (s,α,θk); s is the edge normalized toroidal flux, α is the field line variable, and θk is the perpendicular wave vector or ballooning parameter. Broken symmetry, i.e., deviations from axisymmetry, in the stellarator magnetic field geometry causes localization of the ballooning mode eigenfunction, and gives rise to new types of nonsymmetric eigenvalue isosurfaces in both the stable and unstable spectrum. For eigenvalues far above the marginal point, isosurfaces are topologically spherical, indicative of strong “quantum chaos.” The complexity of QAS marginal isosurfaces suggests that finite Larmor radius stabilization estimates will be difficult and that fully three-dimensional, high-n MHD computations are required to predict the beta limit.

Constraints on the Solar Internal Magnetic Field from a Buoyancy Driven Solar Dynamo

We report here results from a dynamo model developed on the lines of the Babcock-Leighton idea that the poloidal field is generated at the surface of the Sun from the decay of active regions. In this model magnetic buoyancy is handled with a realistic recipe – wherein toroidal flux is made to erupt from the overshoot layer wherever it exceeds a specified critical field B c (105 G). The erupted toroidal field is then acted upon by the α-effect near the surface to give rise to the poloidal field. In this paper we study the effect of buoyancy on the dynamo generated magnetic fields. Specifically, we show that the mechanism of buoyant eruption and the subsequent depletion of the toroidal field inside the overshoot layer, is capable of constraining the magnitude and distribution of the magnetic field there. We also believe that a critical study of this mechanism may give us new information regarding the solar interior and end with an example, where we propose a method for estimating an upper limit of the difusivity within the overshoot layer.

コンパクトトーラス合体実験装置TS-4における同極性および異極性合体の比較実験

For the past fifteen years, the TS-3 merging project has explored several new type of CT/ST(Compact tori and Spherical tokamak): merging formation of FRCs(Field Reversed Configuration), magnetic reconnection experiment, and ultra-high beta ST formation. The successful results of these experiments have enabled us to construct an up-scaled merging device; namely the TS-4. The new device TS-4 has two flux cores(major radius=0.5m) for inductive plasma formation, a center coil(external toroidal field coil and OH coil) for q value control and plasma sustainment, and DC external field coils. The initial operation of the TS-4 (since 2001) demonstrated co-helicity and counter-helicity merging of two spheromaks. The counter-helicity merging revealed annihilation of toroidal magnetic field with a significant overshoot motion of reconnected field lines. It resulted in the formation of an high-beta (≈0.8) FRC. The produced FRC has a significant field deformation, suggesting contribution of toroidal plasma flow to its equilibrium. The co-helicity merging doubled the toroidal fluxes of spheromak, causing its flux conversion from toroidal to poloidal. The merging spheromaks relaxed into another spheromak with beta ≈0, suggesting the large thermal energy loss due to its non-aximetric dynamo activity.

Flux-loss of buoyant ropes interacting with convective flows

We present 3-d numerical magneto-hydrodynamic simulations of a buoyant, twisted magnetic flux rope embedded in a stratied, solar-like model convection zone. The flux rope is given an initial twist such that it neither kinks nor fragments during its ascent. Moreover, its magnetic energy content with respect to convection is chosen so that the flux rope retains its basic geometry while being deflected from a purely vertical ascent by convective flows. The simulations show that magnetic flux is advected away from the core of the flux rope as it interacts with the convection. The results thus support the idea that the amount of toroidal flux stored at or near the bottom of the solar convection zone may currently be underestimated.

Thermal energy confinement of high-triangularity ELMy H-mode plasmas in JT-60U

High-triangularity ELMy H-mode experiments (δx~0.45) have been carried out in JT-60U. In the low-density regime (n̄e/nGW<0.5), the discharges are characterized by type-I ELMs and high-triangularity plasmas produced higher H-mode pedestal pressure than low-triangularity plasmas (δx~0.20). In the high-density regime (n̄e/nGW>0.5), at fixed neutral beam heating power type-III ELMs are generated and the pedestal pressure reduces gradually with density whereas the ELM frequency at low triangularity is much lower at a given density. Compared to low-triangularity plasmas at low densities, higher pedestal temperature was obtained at high triangularity and the core temperature was also improved by almost the same factor. The profiles of temperature are stiff in the sense that there seems to exist a minimum scale length of temperature gradient. Since plasma shaping affects strongly the peripheral region of toroidal magnetic flux surfaces, the improvement of the edge stability by triangularity leads to higher pedestal temperature, which in turn raises the core temperature.

Measurement of plasma diamagnetism in the SINP tokamak by a flux loop system inside the vacuum vessel

Plasma diamagnetism has been measured in the SINP tokamak by a toroidal flux loop placed inside the vacuum vessel. The flux due to the strong toroidal field has been compensated for by a coplaner annular loop which encircles but does not contain the plasma column. The influence of the eddy currents in the vacuum vessel and the conducting shell in these loops has been calculated analytically by a circuit model using the theory of linear networks and compensated accordingly. This method has been shown to yield an almost exact compensation for toroidal flux (∼0.01%) as well as pickups from other fields. Typical results with plasma shots have been presented.

A new paradigm for solar filament eruptions

This article discusses the formation, magnetic structure, and eruption of solar filaments in terms of two contrasting paradigms. The standard paradigm is that filaments are formed by condensation of plasma on coronal magnetic fields that are twisted or dimpled as a result of photospheric motions. According to this paradigm, filaments erupt when photospheric motions shear the fields, increasing their energy and decreasing their stability. According to a new paradigm, subsurface motions generate toroidal magnetic flux ropes, and after these flux ropes emerge to form active regions, the most twisted parts migrate into the corona to form filaments. Filaments become unstable and are ejected after a sufficient accumulation of twist (i.e., magnetic helicity). Various proposed mechanisms for producing the needed helicity are reviewed, and several observational tests are proposed to differentiate among the possible mechanisms.

Magnetic flux evolution in highly shaped plasmas

The resistive evolution of magnetic flux in toroidal devices is studied. The formulation is applicable to general nonaxisymmetric (three-dimensional) toroidal configurations. In particular, it can treat highly shaped, high β, three-dimensional stellarator configurations, as well as two-dimensional (axisymmetric) tokamak plasmas. The time evolution of the poloidal magnetic flux is posed in terms of the rotational transform, ι̸, and allows for a transparent inclusion of stellarator specific current-free contributions to ι̸. Strong diamagnetic and paramagnetic contributions to toroidal magnetic flux, as evident in spherical tokamaks and similar concepts, are calculated by direct iteration with an equilibrium solver. The nonlinear evolution equation is derived using a susceptance matrix formulation originally introduced by Grad and co-workers [Bateman, Nucl. Fusion 13, 227 (1973)]. Here, it is extended to general, nonstraight field line coordinate systems. The basic equations are described, explicit expressions for the susceptance matrix are given, and example applications using the stand-alone code, THRIFT (THRee-dimensional Inductive Flux evolution in Toroidal devices), are discussed.

Analytical models of axisymmetric toroidal magnetic fields withnon-circular flux surfaces

Axisymmetric toroidal magnetic configurations relevant to current andfuture tokamak experiments are analytically modelled taking into account anarbitrary aspect ratio, noncircularity and up-down asymmetry of fluxsurface shapes. It is shown that equilibrium tokamak magnetic fields can besatisfactorily described using only four geometric parameters, whichdetermine the shape and position of the flux surfaces. These are theShafranov shift, the elongation, and the parameters of triangularity and ofup-down asymmetry. To cover the variety of observed flux surface shapes,three different analytical expansions are provided. Comparison of themodelled with the equilibrium flux surfaces in NSTX (Princeton), JET(Abingdon) and TCV (Lausanne) demonstrates good coincidence.

Formation and sustainment of electrostatically driven spheromaks in the resistive magnetohydrodynamic model

The nonlinear time-dependent equations of resistive magnetohydrodynamics are solved in simply connected domains to investigate spheromak formation and sustainment with electrostatic current drive. Spheromak magnetic fields are generated in three-dimensional computations as the nonlinear state resulting from an unstable pinch. Perturbations convert continuously supplied toroidal magnetic flux into poloidal magnetic flux, leading to “flux amplification” of field embedded in the electrodes. Relaxation of the axisymmetric component of the parallel current profile can be substantial, and the final nonlinear state is steady over a wide range of parameters. However, for sufficiently large values of Lundquist number or sufficiently large applied potential, nonsteady final states are observed with periodic relaxation events in some cases. Under most conditions, the saturated configuration exhibits chaotic scattering of the magnetic field lines. Conditions just above the marginal point of pinch instability sustain large closed flux surfaces in steady state; a weakly kinked pinch current threads the toroidal region of closed flux surfaces and imposes stellarator-like helical transform. Closed flux surfaces also form during decay, due to reduced fluctuation levels and average toroidal current driven directly by inductive electric field.

November 17–18, 1975, event: A clue to an internal structure of magnetic clouds?

During November 17–18, 1975, a magnetic cloud was observed by the IMP 8 satellite. The cloud was analyzed in several papers. It draws attention because it is the most clear example where the magnetic field components behave differently from the current single flux rope model. Various models and fits have been presented to explain the magnetic field measurements for this particular event: single‐polarity cylindrical flux rope, spheromak, toroidal flux rope, and two subsequent flux ropes (flux rope twins). We critically examine these models and fits and stress that not only magnetic field data but also plasma data must be taken into account. There is a remarkably sharp drop in the density inside the magnetic cloud. The most consistent explanation of the behavior of magnetic field and plasma data for this event is that the magnetic cloud consists of a dual‐polarity flux rope with a low density and strong magnetic field core surrounded by an annular region of the same chirality but opposite polarity. An implication of this possibility to explain other magnetic cloud observations is discussed.

M = 0 perturbations of the magnetic surfaces in an RFP

An analysis of the m = 0 magnetic perturbation in the vacuum region of anRFP device is carried out using toroidal flux and toroidal fieldmeasurements. This perturbation is associated to the locked-mode phenomenonwhich is a coherent superimposition of modes, mainly m = 0,1, locked inphase together. The m = 1 distortion of the magnetic surface is thedominant effect and was considered in a previous analysis. The purpose ofthis work is to study the m = 0 component using the same technique. Theexperimental data are elaborated with simple analytical expressions derivedfrom the vacuum Maxwell equations and the general contravariantrepresentation of the magnetic field in terms of the flux function. Them = 0 geometrical distortion of the magnetic surfaces is reconstructed. Itsdependence on the plasma current and the impact on the plasma wallinteraction are discussed.

Theory for toroidal momentum pinch and flow reversal in tokamaks.

It is demonstrated that besides the well-known toroidal momentum diffusion flux there is a pinchlike flux in the fluctuation-induced toroidal stress. A toroidal flow profile is determined up to a constant, e.g., the value of the flow at the magnetic axis, by balancing these two fluxes. The remaining residual toroidal stress determines the value of the flow at the axis. It is illustrated that the direction of the flow at the axis can change after plasma confinement is improved. The theory is applied to explain the toroidal flow reversal in tokamak experiments.

Sunspot groups as tracers of sub-surface processes

Data on sunspot groups have been quite useful for obtaining clues to several processes on global and local scales within the sun which lead to emergence of toroidal magnetic flux above the sun&#x2019;s surface. I present here a report on such studies carried out at Indian Institute of Astrophysics during the last decade or so.