Bending Of Field Lines Research Articles

Two-dimensional modelling of blob dynamics in tokamak edge plasmas

Dynamics of blobs described by two different models for the closures of parallel plasma current is studied numerically. One of these closures describes the blob sitting on magnetic lines going in the scrape off layer through the conducting surface, while the other one describes the blob with high plasma pressure where the bending of magnetic field lines becomes important.

Three‐dimensional magnetotail equilibria with prescribed boundary shapes

Three‐dimensional magnetotail equilibrium configurations are presented that are derived from a prescribed outer flux surface, equivalent to a closed magnetopause. The explicit solutions neglect the effects of field‐aligned currents, closely associated with a bending of field lines in the x, y plane, but include tail flaring in y and z. Most of the solutions are derived for an axisymmetric boundary, approximately consistent with force balance between the unperturbed solar wind and the magnetosphere at a closed magnetopause (Newtonian approximation). This force balance is used to relate the downtail variation of the tail radius to that of the total interior magnetotail pressure, equivalent to the lobe magnetic pressure. However, these properties are not sufficient to determine the tail configuration. We demonstrate this by deriving solutions with identical boundaries and identical variation of the total pressure along the tail but different amounts of field line flaring in the x, y plane. Under these conditions, stronger field line flaring is associated with a reduction of the magnetotail plasma pressure near midnight and an increase of the plasma sheet thickness from midnight toward the tail flanks. A reduction of the tail boundary flaring leads to an increase of the plasma sheet thickness near the Earth but does not significantly affect the cross‐tail variation. Nonuniform boundary flaring with a flattening of the tail, corresponding to a more significant increase of the tail radius in y, can reduce the plasma sheet thickening from midnight to the tail flanks or even reverse it to a thinning. The explicit solutions illustrate the effects on the tail structure of both the instantaneous force balance between the magnetosphere and the solar wind and the history of this interaction, which affects the plasma and magnetic flux distribution inside the magnetotail.

Polarized dust emission of magnetized molecular cloud cores

We compute polarization maps for molecular cloud cores modeled as magnetized singular isothermal toroids, under the assumption that the emitting dust grains are aspherical and aligned with the large-scale magnetic field. We show that, depending on the inclination of the toroid with the line-of-sight, the bending of the magnetic field lines resulting from the need to counteract the inward pull of gravity naturally produces a depolarization effect toward the centre of the map. We compute the decrease of polarization degree with increasing intensity for different viewing angles and frequencies, and we show that an outward increasing temperature gradient, as expected in starless cores heated by the external radiation field, enhances the decrease of polarization. We compare our results with recent observations, and we conclude that this geometrical effect, together with other mechanisms of depolarization, may significantly contribute to the decrease of polarization degrees with intensity observed in the majority of molecular cloud cores. Finally, we consider the dependence of the polarization degree on the dust temperature gradient predicted for externally heated clouds, and we briefly comment on the limits of the Chandrasekhar-Fermi formula to estimate the magnetic field strength in molecular cloud cores.

Relation of polar auroral arcs to magnetotail twisting and IMF rotation: a systematic MHD simulation study

Abstract. We investigate with the help of a magnetohydrodynamic (MHD) model how the large-scale topology of the magnetosphere develops for a constant interplanetary magnetic field (IMF) with different IMF clock angles and for an IMF By sign change during northward IMF. A detailed examination of the topological changes in the tail and the ionosphere for different IMF conditions shows a good agreement with observational results. The MHD simulations for different constant IMF clock angle cases show the expected field-line bending and tail twisting for nonzero IMF By. The tail becomes longer and at its tailward end stronger twisted for IMF Bz>∣By∣ than for IMF Bz

Controlled alignment of carbon nanofibers in a large-scale synthesis process

Controlled alignment of catalytically grown carbon nanofibers (CNFs) at a variable angle to the substrate during a plasma-enhanced chemical vapor deposition process is achieved. The CNF alignment is controlled by the direction of the electric field lines during the synthesis process. Off normal CNF orientations are achieved by positioning the sample in the vicinity of geometrical features of the sample holder, where bending of the electric field lines occurs. The controlled growth of kinked CNFs that consist of two parts aligned at different angles to the substrate normal also is demonstrated.

Importance of Finite Thermal Energy Density and Singular Modes for the Transport of Angular Momentum in Accretion Disks

In thin accretion disks the electromagnetic modes that can transport angular momentum (in the outward direction as needed for accretion to occur) are shown to require, for their excitation, the combined effects of the finite plasma temperature and of the gradient of the rotation frequency Ω(R) in the realistic case where the disk magnetic field has a toroidal component. In particular, the contribution of finite plasma compressibility (▿ · v≠0) is shown to play a key role. Accordingly, the relevant instabilities would not grow out of a cold plasma disk and pre-heating becomes necessary for the occurrence of momentum transport and accretion. The new kind of modes that are found are singular when described by the ideal MHD approximation, corotate with the plasma at the radius where their amplitude is maximum, are non-axisymmetric, and are strongly affected by the gradient of the relevant Doppler shifted frequency ω=ω−n0Ω (n0= toroidal mode number ≠0). The case where a toroidal and a vertical component of the magnetic field are present is considered. Then the minimum value of the parallel, to the field, component of the relevant wave vector is not bound by the height of the disk and the bending of the magnetic field lines by the modes can be optimized to minimize the instability threshold. The ideal MHD equation for the radial amplitude is characterized by two singularities: one corresponds to the radii where ω equals the “slow” magnetosonic mode frequency, that vanishes if the plasma temperature vanishes, and the other to the radii where ω equals the shear-Alfvén frequency. The key role is played by the first kind of resonance and the presence of a very small and locally enhanced rate of dissipation (e.g., finite thermal conductivity) is required to “remove” the relevant singularities and produce meaningful growth rates. By contrast, axisymmetric (n0=0) modes, that are shown to acquire a vertical ballooning structure, in a thin disk, are considerably more difficult to excite. The importance of finite compressibility is demonstrated also for this kind of modes.

Effect of toroidal current and profile on ballooning instabilitymechanisms in a quasiaxisymmetric stellarator

Finite toroidal plasma current reduces the magnitude of the normal magneticfield line curvature near midvolume in a quasiaxisymmetric stellarator.This improves the ideal ballooning stability conditions.Near the edge of the plasma, a hollow toroidal current profile deterioratesthe ballooning stability properties by enhancing the normal curvature, whilea parabolic current profile improves this stability because the normalcurvature decreases. The magnetic field line bending term can alsoimpact the stability properties when the normal curvatures are comparable.The local magnetic shear and the geodesic curvature vanish where the normalcurvature is most destabilizing. The quasiperiodic ballooning curvaturecontribution, unlike the normal curvature, fails to constitute a usefulindicator about the localization of ballooning structures.

Nonlinear Growth of the Three‐dimensional Undular Instability of a Horizontal Magnetic Layer and the Formation of Arching Flux Tubes

We use an anelastic MHD code to simulate the nonlinear evolution of the three-dimensional undular instability of a horizontal magnetic layer with a fixed field line direction, embedded in an adiabatically stratified atmosphere. We consider the limit of very high plasma β, representing the condition at the base of the solar convection zone. We show that, in the limit of high plasma β and nearly adiabatic stratification, the anelastic formulation gives an accurate description of the magnetic buoyancy instabilities. We specify the thermodynamic conditions of the magnetic layer such that it is stable against pure interchange modes (with zero wavenumber in the direction of the magnetic field) and is unstable only to three-dimensional undular modes (with nonzero wavenumbers in both horizontal directions parallel and perpendicular to the field). Our simulations show that distinct arching flux tubes form as a result of the growth of the three-dimensional undular instability. The apices of the arching tubes become increasingly buoyant because of the diverging mass flow from the apices to the troughs. The field strength at each loop apex decreases with height at a significantly smaller rate in comparison with that for the rise of a horizontal flux tube, because of the stretching of the loop field lines. Even though the initial magnetic field is untwisted, it is found that the upward moving tube cross sections of the arching tubes maintain their cohesion as they rise through the distance of about 1 density scale height included in the simulation domain. The difference in motion between the apices and the troughs causes bending and braiding of the longitudinal field lines, whose restoring tension force improves the cohesion of the rising flux tubes in comparison with previous two-dimensional simulations of the buoyant rise of horizontal flux tubes with no initial twist. In addition, the fact that both the buoyancy and the tension forces grow self-consistently from zero as the tubes arch is also a crucial factor for the cohesion of the rising tubes. The result of our simulations suggests that the minimum value for the ratio of poloidal field strength over toroidal field strength (i.e., twist) at the base of the solar convection zone, necessary to ensure a cohesive rise of magnetic flux through the solar convection zone, may be far less than that suggested by the two-dimensional calculations of the buoyant rise of infinitely long horizontal tubes.

Theory of nonoverlapping low frequency modes in axisymmetric toroidal plasmas

The theory of low frequency instabilities in axisymmetric toroidal plasmas is presented from the point of view of the two-fluids equations, assuming the standard drift wave ordering. Attention is focused on the limit in which neighboring rational surfaces (corresponding to a fixed toroidal mode number) are sufficiently far apart that mode overlapping can be neglected. Owing to field line bending, poloidal side bands m±1,… coexist with the primary mode m, enhancing noticeably the role of the parallel ion dynamics. The electron and ion branches are investigated systematically under those conditions. It is found that the radial widths of the eigenmodes increase with respect to the slab values; the shear damping rate of the electron branch, respectively, the growth rate of the ion branch increase correspondingly. Other interesting new results are obtained concerning, in particular, the frequency, the growth rate and the poloidal asymmetry of the ion mode fluctuations. It is mentioned that those appear to be directly relevant to various experiments and, in particular, to internal transport barriers. Shortcomings of the standard ballooning formalism are pointed out.

Vertical stability in a current-carrying stellarator

An analytic stability criterion is derived for the vertical mode in a large aspect ratio stellarator with uniform current density profile. The effects of vacuum magnetic field generated by helical coils are shown to be stabilizing due to enhancement of field line bending energy. For a wall at infinite distance from the plasma, the amount of external poloidal flux needed for stabilization is given by f=(κ2−κ)/(κ2+1), where κ is the axisymmetric elongation and f is the ratio of vacuum rotational transform to the total transform.

Minimal field bending modes in hot accretion disks

In a configuration that can simulate hot accretion disks where both a toroidal and a poloidal component of the magnetic field are present, a new class of modes is found that minimizes the bending of the magnetic field lines and involves a novel effect related to (the square of) the rotation frequency gradient. These modes, which co-rotate with the plasma at the radius around which they are localized and depend explicitly on the radial width of localization, have the lowest threshold among the various types of mode that are possible as the parallel, to the field, component of the relevant mode number is decoupled from the magnitude of the height of the disk. The new effect and driving factor is related to the gradient of the Doppler shifted frequency ω − n 0Ω(R) , where Ω(R) is the rotation frequency, n 0 the toroidal mode number and ω the mode frequency. The contribution of a previously known driving factor, identified first by Velikhov and associated with the linear contribution of dΩ dR to the perturbed toroidal field, remains important. These modes that depend on the plasma temperature being finite, survive the inclusion of relevant dissipative effects and can be expected to persist in collisionless regimes where the adopted fluid description is to be reconsidered.

The IMF bending upstream of the Martian bow shock

A noticeable bending of the magnetic field lines in the Martian upstream region is observed. Variations of the field are controlled by the IMF and the foreshock geometry. The sign of ΔB changes at the crossing of the tangent surface, which borders foreshock and the undisturbed solar wind. Perturbations of the magnetic field are centered at the plane which is normal to the tangent surface and intersects the bow shock at the tangent point. All observed signatures suggest that the current, providing field variations, is localized near the tangent surface. Although, two models are discussed, neither of them is able to explain all observed features satisfactorily.

Interaction of Planetary Nebulae with a Magnetized ISM

We study the interaction of planetary nebulae with a magnetized interstellar medium (ISM) and show that the ISM magnetic field cannot be ignored in analysis of the interaction of planetary nebulae with (PNs) with the ISM. We show that stretching and bending of magnetic field lines can increase the effective pressure of the ISM. In particular, near the stagnation point of the flow of the ISM around the nebula, the ISM magnetic pressure can overcome the thermal pressure, leading to strong MHD instabilities even in isothermal flows. At early stages, when the planetary nebula outer regions are still neutral, the ISM magnetic field can penetrate the nebula via ambipolar diffusion. As a result, when pulled downstream by the flow, the magnetic field will strip the outer layers of the nebula. The Rayleigh-Taylor and Kelvin-Helmholtz instabilities with wavenumbers along the magnetic field lines are suppressed. Therefore, the ISM magnetic field can break the cylindrical symmetrical nature of the interaction.

Ideal Current Layers and Kink Instability in Line-Tied Coronal Loops

The development of the kink instability in line-tied coronal loops is studied using a cylindrical MHD code. When the twist of magnetic field lines of the initial configuration exceeds a critical value, an ideal kink mode develops and drives this unstable equilibrium towards a secondary bifurcated equilibrium containing an electric current concentration. Contrary to a periodic untied configuration where a current sheet is ideally generated, the current layer is non-singular with a non-zero thickness and a finite amplitude. This current concentration extends along all the loop length and takes the form of an helical ribbon of intense current. The numerical results give an algebraic linear-like scaling of the characteristics of the current layer (amplitude and thickness) as a function of the aspect ratio of the loop. An interpretation in terms of axial field-line bending of the three-dimensional kinked equilibrium is proposed.

High-mode-number ballooning modes in a heliotron/torsatron system. II. Stability

As described in the companion paper [N. Nakajima, Phys. Plasmas 3, 4545 (1996)], in heliotron/torsatron systems that have a large Shafranov shift, the local magnetic shear is found to have no stabilizing effect on high-mode-number ballooning modes at the outer side of the torus, even in the region where the global shear is stellarator-like in nature. The disappearance of this stabilization, in combination with the compression of the flux surfaces at the outer side of the torus, leads at relatively low values of the plasma pressure to significant modifications of the stabilizing effect due to magnetic field-line bending on high-mode-number ballooning modes—specifically, that the field-line bending stabilization can be remarkably suppressed or enhanced. In an equilibrium that is slightly Mercier-unstable or completely Mercier-stable due to peaked pressure profiles, such as those used in standard stability calculations, high-mode-number ballooning modes are destabilized due to these modified stability effects, with their eigenfunctions highly localized along the field line. Highly localized mode structures such as these cause the ballooning mode eigenvalues ω2 to have a strong field line dependence (i.e., α-variation) through the strong dependence of the local magnetic curvature, such that the level surfaces of ω2(ψ,θk,α) (≤0) become spheroids in (ψ,θk,α) space, where ψ labels flux surfaces and θk is the radial wave number. Because the spheroidal level surfaces for unstable eigenvalues are surrounded by level surfaces for stable eigenvalues of high-mode-number toroidal Alfvén eigenmodes, those high-mode-number ballooning modes never lead to low-mode-number modes. In configuration space, these high-mode-number modes are localized in a single toroidal pitch of the helical coils, and hence they may experience substantial stabilization due to finite Larmor radius effects.

Model of the low‐latitude boundary layer with finite field‐aligned potential drops and nonconstant mapping factors

A 2½‐dimensional, steady state model of the low‐latitude boundary layer (LLBL) on closed field lines has been developed. Coupling to the dayside auroral ionosphere via region 1 field‐aligned currents and associated finite field‐aligned potential drops is included. The boundary layer (BL) flow is influenced by viscous forces, inertia forces, pressure forces, and j × B forces. The BL approximation and kinematic treatment of the plasma motion perpendicular to the equatorial plane greatly simplify the model. The model has three modules: an equatorial LLBL module, a force‐free coupling module, and an ionospheric module. It includes nonconstant magnetosphere‐ionosphere mapping factors, describing the increasing stretching of magnetic field lines with increasing distance away from the subsolar region, as well as variations in mapping factors across the BL. In this paper, both the BL height and the ionospheric conductivities are assumed constant. The effects of finite field‐aligned potential drops and nonconstant mapping factors on the BL flow, on the plasma density and temperature, on the magnetic field line bending, and on the field‐aligned current distribution are presented. The effects of varying the magnitude of the (constant) ionospheric Pedersen conductivity are also discussed. In addition, the effects of viscosity, including varying its magnitude and allowing it to be a function of location both along and across the LLBL, are examined. A number of qualitative features of the simulation results are found to be consistent with various observations.

An analytic study of the magnetohydrodynamic stability of inverse shear profiles

This paper reports on the ideal magnetohydrodynamic (MHD) stability of tokamak field profiles that have a non-monotonic safety factor q(r). An analytic criterion is obtained for these ‘‘inverse shear’’ profiles by expanding in inverse aspect ratio and assuming that the minimum in q is slightly less than the m/n value of the mode under examination (m and n being the principal poloidal and toroidal mode numbers of the instability). Three terms are identified as controlling the stability of this ‘‘double kink’’; two of them are stabilizing and due, respectively, to field line bending and the interaction of average favorable curvature with the pressure gradient. The possibility of instability comes from the third term which is due to toroidal coupling and is ballooning in character. The analytic results are compared with those from a fully toroidal stability code.

On the possibility of the MHD‐ballooning instability in the magnetotail‐like field reversal.

A stability analysis of the antisymmetric ballooning mode is presented for the tail like magnetic models within ideal MHD. The analysis particularly emphasizes the potential importance of the field‐aligned variation of the local field curvature. First, it is found that the tail field models of Kan [1973] are stable against the antisymmetric ballooning mode because of the strong stabilizing effect of the field line bending. In such models, the field curvature is significant only in the narrowly limited region, namely, very near the equatorial plane from which it then rapidly declines, more rapidly than 1/B, along the field line. Then, the antisymmetric ballooning mode is studied in a magnetic field model in which the field curvature is assumed to vary as 1/B over some small portion from the equatorial point along the field line. For such a model, two criteria for the instability are proposed, one as a necessary condition and the other as a sufficient condition. Not only a sufficient pressure gradient and/or a high‐plasma beta, but also a sufficient field‐aligned portion of a substantial curvature, seem necessary for a tail‐like field line to become unstable to the antisymmetric ballooning mode. As a specific example, it is shown that a high‐beta, tail field line can suffer from an antisymmetric ballooning instability if the field line has a geometrical roundness over a distance of Rce the equatorial radius of the field curvature, from the equatorial position.

Anisotropic Alfvén‐ballooning modes in Earth's magnetosphere

We have carried out a theoretical analysis of the stability and parallel structure of coupled shear Alfvén and slow magnetosonic waves in Earth's inner magnetosphere (i.e., at equatorial distances between about five and ten Earth radii) including effects of finite anisotropic plasma pressure. Multiscale perturbation analysis of the anisotropic Grad‐Shafranov equation yields an approximate self‐consistent magnetohydrodynamic (MHD) equilibrium. This MHD equilibrium is used in the numerical solution of a set of eigenmode equations which describe the field line eigenfrequency, linear stability, and parallel eigenmode structure. We call these modes anisotropic Alfvén‐ballooning modes. The main results are: (1) The field line eigenfrequency can be significantly lowered by finite pressure effects. (2) The parallel mode structure of the transverse wave components is fairly insensitive to changes in the plasma pressure, but the compressional magnetic component can become highly peaked near the magnetic equator as a result of increased pressure, especially when P⊥ > P∥ (here P⊥ and P∥ are the perpendicular and parallel plasma pressure). (3) For the isotropic (P∥ = P⊥ = P) case ballooning instability can occur when the ratio of the plasma pressure to the magnetic pressure, β = P/(B²/8π), exceeds a critical value β0B ≈ 3.5 at the equator. (4) Compared to the isotropic case the critical beta value is lowered by anisotropy, either due to decreased field line bending stabilization when P∥ > P⊥ or due to increased ballooning‐mirror destabilization when P⊥ > P∥ (5) We use a β‐δ stability diagram to display the regions of instability with respect to the equatorial values of the parameters β and δ, where β¯=(1/3)(β∥+2β⊥) is an average beta value and δ = 1 − P∥/P⊥ is a measure of the plasma anisotropy. The diagram is divided into regions corresponding to the firehose, mirror and ballooning instabilities. It appears that observed values of the plasma pressure are below the critical value for the isotropic ballooning instability but it may be possible to approach a ballooning‐mirror instability when P⊥/P∥ ≳ 2.

MHD waves in coronal flux tubes

The basic MHD waves of a coronal flux loop are investigated for the rectangular box model of a plasma with oblique magnetic field and line-tied at the ends. The waves found are completely different from those in a periodic box, representative for tokamaks. They consist of a mixture of Alfven components with a ballooning factor, favouring minimal field line bending, and fast components without such a factor. Pure Alfven modes are only found as singular limiting cases of cluster spectra of Alfven-fast waves, where the fast components are localised in a photospheric boundary layer which is dictated by the requirements of line-tying. This justifies the assumption of continuous spectra in coronal loops, required for the mechanism of resonant Alfven wave heating. The waves consist of large amplitude Alfven components in the corona and fast components with a small but rapidly varying amplitude in the boundary layer, so that they appear to have the right signature for effective transfer of energy from the photosphere to the corona.