Equilibrium of magnetic fields with arbitrary interweaving of the lines of force i. discontinuities in the torsion

Abstract

This paper considers the static force-free equilibrium V×B=αB of a magnetic field in which all of the lines of force connect without knotting between parallel planes. The field is formed by continuous deformation from an initial uniform field, and is conveniently described in terms of the scalar function ψ, which is effectively the stream function for the incompressible wrapping and interweaving of the lines of force, and the scalar function θ, which describes the local compression and expansion. Equilibrium requires satisfaction of two independent equations (the third equation defines α), which cannot be accomplished without the full freedom of both functions ψ and θ. It is shown by integration along the characteristics of the equilibrium equations that, when ψ is predetermined by an arbitrary winding pattern, there appear discontinuities in α. Discontinuities in α have discontinuities in the field (i.e. current sheets) associated with them.

We expect such discontinuities to be produced in the magnetic fields extending outward from the convecting surfaces of the cooler stars.     

Tangential discontinuities and the optical analogy for stationary fields IV. High speed fluid sheets

Abstract

It was shown in the previous paper that a sufficiently strong pressure maximum applied to an equilibrium flux surface, by the fields on either side of the surface, produces a gap in the flux surface. The fields on either side make contact through the gap to produce a surface of tangential discontinuity (current sheet). It is shown in the present paper that there is a high speed sheet of fluid and field sliding over the surface of discontinuity when the applied pressure moves slowly across the flux surface. Conditions in the active X-ray corona of the sun suggest that such sheets are generally present, with velocities of the order of 10² km/sec, but with thicknesses too small to be observed. More substantial high speed sheets of fluid may occur in solar flares.     

Tangential discontinuities and the optical analogy for stationary fields. v. formal integration of the force-free field equations

Abstract

This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ? x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field B_x = + B ₀cosqz, B_y = — B ₀sinqz, B_z = 0, anchored at the distant cylindrical surface ? = (x ² + y ²)^1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l ²/R ²)]. The field equations can be integrated over ? = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ? = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape this restriction only by developing internal discontinuities.

Casting the field equation in a form that the connectivity can be specified explicitly, reduces the field equation to the eikonal equation, describing the optical analogy, treated in papers II and III of this series. This demonstrates the ubiquitous nature of the tangential discontinuity in a force-free field subject to any local deformation.     

Tangential discontinuities and the optical analogy for stationary fields. VII. Effects of resistivity

Abstract

The relatively large resistivity in the solar photosphere and chromosphere softens the ideal tangential discontinuities of magnetostatic equilibrium into continuous transitions in field direction over scales of 0.1–10 km. This softening is communicated upward at the Alfvén speed into the active solar corona. The degree of softening is a vital part of the theory of magnetic heat input to the active X-ray corona, because the very low resistivity of the coronal gas provides effective dissipation only if the current sheets are reduced to a thickness of 10^?2km.

A close examination of the problem shows that the Alfvén transit time up into the corona is large compared to the characteristic time of 1 sec in which the coronal tangential discontinutities are formed. It also shows that the principal effect of the resistivity is to create a thin surface layer of fluid on adjacent flux bundles, which causes a general drift of the flux but does not directly broaden the current sheets higher up in the field. In fact the motions of the surface layers do not extend upwards beyond the first winding pattern at each end of a coronal loop.

It appears that the photospheric and chromospheric resistivity is without striking consequences for magnetic heating in the corona.     

Plane discontinuities in collisionless plasmas: An analysis as applied to the magnetopause

Types of plane discontinuities possible in collisionless kinetics—thin current sheets with a nonzero normal component of the magnetic field—are considered. In a general case the electromagnetic field energy is transformed at such a discontinuity to the energy of ion flows. In addition to the known MHD discontinuities, the discontinuities that exist in the presence of pairs of interpenetrating flows on both sides of a discontinuity are studied. The application of the theory to an analysis of observations on the magnetopause has been considered.     

Surface waves on MHD tangential discontinuities: case for moderate to low plasma-β values

Variation of phase speed of the magnetoacoustic surface waves on a magnetohydrodynamic (MHD), tangential discontinuity (TD) is examined against the variation of Alfvén speeds on opposite sides of the discontinuity with the acoustic speeds remaining constant. The discontinuous thermal profile of the plasma is employed to classify various surface wavemodes. The existence conditions for the surface waves manifest themselves through the appearance of critical values for various parameters restricting the propagation of such waves on the TD. The limiting situation of extremely low-β plasmas is studied to improve upon earlier results based on simplified calculations. Theoretical understanding of the basic structure of surface waves is then applied to specific observations of discontinuities in the interplanetary magnetic field (IMF) to arrive at simple estimates of phase speed of the surface waves on such discontinuities that may be tested in further observations.     

Ion--neutral friction as a model for magnetic relaxation

It is generally admitted that a plasma in the absence of forcing will relax to a minimum energy state compatible with appropriate constraints. Usually this is a force-free state, which, in two dimensions, implies a potential magnetic field except by the possible presence of current sheets. The precise mechanism of this relaxation, and in particular the plasma velocity, are generally ignored. There exists, however, a physically well-defined process that should produce magnetic relaxation: ion--neutral (or ambipolar) friction. While there is no guarantee of the existence of a limit of this process when t?→?∞, there exists a family of sequential limits for whom the Lorentz force tends to zero. To analyze the configuration of these limit states, we study the evolution of several moments of the magnetic energy. We prove that for as long as the enstrophy remains bounded, the current density energy also remains bounded in two dimensions: this excludes all classical configurations of current sheets across which the magnetic field reverses direction. Hence, these sheets cannot be the limit of ion--neutral diffusion unless the flow becomes increasingly irregular over time.     

On the magnetic flux linkage of an electrically-conducting fluid

Abstract

A method has recently been proposed for finding the radius r_c of the electrically-conducting fluid core of a planet of outer radius r_s from observations of the magnetic field B in the accessible region near or well above the surface of the planet (Hide, 1978). The method is based on the supposition that when the magnetic field is produced by hydromagnetic dynamo action in the core, implying that the magnetic Reynolds number R there is large, (a) fluctuations in B can occur everywhere on the comparatively short advective time-scale τ _A associated with fluid motions in the core and so can fluctuations in the quantity N, defined for any closed surface S as the total number of intersection of magnetic lines of force with S, provided that S lies well outside the core, but (b) at the surface of the core, where lines of magnetic force emerge from their region of origin, concomitant fluctuations in N are negligibly small, of the order of τ _A/τ_O where τ _O (= Rτ_A ) is the Ohmic decay time of the core.

A proof of this supposition follows directly from the general expression derived in the present paper showing that when S is a material surface the time rate of change of N is equal to minus twice the line integral of the current density divided by the electrical conductivity around all the lines on S where the magnetic field is tangential to S. This expression (which Palmer in an accompanying paper rederives and extends to the relativistic case using the mathematical formalism of Cartan’s exterior calculus) also provides a direct demonstration of the well-known result that although high electrical conductivity, sufficient to make R ? 1, is a necessary condition for hydromagnetic dynamo action, such action is impossible in a perfect conductor, when R→ ∞.     

Wave-induced drift of large floating sheets

In this article we study the wave-induced drift of large, flexible shallow floating objects, referred to as sheets. When surface waves propagate through a sheet, they provide a mean stress on the sheet, resulting in a mean drift. In response, the sheet generates an Ekman current. The drift velocity of the sheet is determined by (i) the wave-induced stress, (ii) the viscous stress due to the Ekman current, and (iii) the Coriolis force. The sheet velocity and the current beneath the sheet are determined for constant and depth-varying eddy viscosities.     

Global magnetohydrostatic fields in stellar atmosphere

Abstract

The equilibrium properties of the magnetic field of an axisymmetric star are studied. A family of analytical solutions to the magnetohydrostatic equations is found, which are used to model the slow evolution of the field through a series of equilibria.

Firstly, a model is set up for a force-free dipole-like field, which has a toroidal field component; it is found that, as such a field is twisted up, a critical point is reached, at which the field topology changes. If the twist is increased beyond this point, there is no physically reasonable equilibrium. Next, an untwisted magnetostatic dipole-like field is studied, with an increasing pressure differential between pole and equator. A critical point again occurs when the pressure differential becomes too large. Finally a force-free quadrupole-like field is modelled, which is being twisted up, for example by differential rotation; this has similar properties to the dipole-like field. In each case, it is suggested that, when the critical point is reached, the field will no longer evolve smoothly, but will change catastrophically to a new stable, releasing energy. Such an event could represent the onset of a stellar flare or some other dynamic stellar process.     

Emplacement and arrest of sheets and dykes in central volcanoes

Sheet intrusions are of two main types: local inclined (cone) sheets and regional dykes. In Iceland, the inclined sheets form dense swarms of (mostly) basaltic, 0.5–1 m thick sheets, dipping either at 20–50° or at 75–90° towards the central volcano to which they belong. The regional dykes are (mostly) basaltic, 4–6 m thick, subvertical, subparallel and form swarms, less dense than those of the sheets but tens of kilometres long, in the parts of the volcanic systems that are outside the central volcanoes. In both types of swarms, the intrusion intensity decreases with altitude in the lava pile. Theoretical models generally indicate very high crack-tip stresses for propagating dykes and sheets. Nevertheless, most of these intrusions become arrested at various crustal depths and never reach the surface to supply magma to volcanic eruptions. Two principal mechanisms are proposed to explain arrest of dykes and sheets. One is the generation of stress barriers, that is, layers with local stresses unfavourable for the intrusion propagation. The other is mechanical anisotropy whereby sheet intrusions become arrested at discontinuities. Stress barriers may develop in several ways. First, analytical solutions for a homogeneous and isotropic crust show that the intensity of the tensile stress associated with a pressured magma chamber falls off rapidly with distance from the chamber. Thus, while dyke and sheet injection in the vicinity of a chamber may be favoured, dyke and sheet arrest is encouraged in layers (stress barriers) at a certain distance from the chamber. Second, boundary-element models for magma chambers in a mechanically layered crust indicate abrupt changes in tensile stresses between layers of contrasting Young’s moduli (stiffnesses). Thus, where soft pyroclastic layers alternate with stiff lava flows, as in many volcanoes, sheet and dyke arrest is encouraged. Abrupt changes in stiffness between layers are commonly associated with weak and partly open contacts and other discontinuities. It follows that stress barriers and discontinuities commonly operate together as mechanisms of dyke and sheet arrest in central volcanoes.     

Studying ice sheet stability using the method of separation of variables

Abstract

Stability of two-dimensional stationary continental and marine ice sheets is studied using perturbations of ice sheet surface elevation and the margin position about a steady-state solution. Ice accumulation rate at the upper surface is specified as a function of elevation and span. Ice is considered as a Newtonian fluid. Linearisation and separation of variables yield a non-homogeneous eigenvalue problem. In case when the margin moves, a necessary condition for the existence of a solution is very restrictive on the functions of the bed profile and accumulation rate. The method of separation of variables is applicable when the margin is assumed to be stationary. Singularity of the perturbation at the margin is treated using the method of matched asymptotic expansions. Numerical experiments show that dependence of accumulation rate on elevation contributes strongly towards ice sheet instability. Bed slope, ice thickness at the grounding line of a marine ice sheet and equilibrium line inclination are the main parameters determining the ice sheet reaction to a surface perturbation.     

非稳态多重X线重联的类型与特征

为了进一步认识间歇性多重Ｘ线重联的特征，本文分析计算结果，发现入流自边界进入后向两侧发散，沿着ｘ轴（ｚ＝０）和出流边界（ｚ＝２），压强Ｐ和磁场Ｂ的分布形态表明这种非稳态重联属快模式扩张型．数值结果还表明，进入扩散区的入流马赫数Ｍ_ｉ与磁雷诺数Ｒｍ之间基本满足关系式Ｍ_ｉ≈１．５Ｒ_ｍ^－１／２─１．７５Ｒ_ｍ^－１／２，而且等离子体的加速与压强Ｐ_ｔ(Ｐ_ｔ     

Impact of textural patterns on modeled rock weathering rates and size distribution of weathered grains

Rock texture has a critical influence on the way rocks weather. The most important textural factors affecting weathering are grain size and the presence of cracks and stylolites. These discontinuities operate as planes of mechanical weakness at which chemical weathering is enhanced. However, it is unclear how different rock textures impact weathering rates and the size of weathered grains. Here, we use a numerical model to simulate weathering of rocks possessing grain boundaries, cracks, and stylolites. We ran simulations with either synthetic or natural patterns of discontinuities. We found that for all patterns, weathering rates increase with discontinuity density. When the density was <~25%, the weathering rate of synthetic patterns followed the order: grid > honeycomb > Voronoi > brick wall. For higher values, all weathering rates were similar. We also found that weathering rates decreased as the tortuosity of the pattern increased. Moreover, we show that textural patterns strongly impact the size distributions of detached grains. Rocks with an initial monomodal grain size distribution produce weathered fragments that are normally distributed. In contrast, rocks with an initial log-normal size distribution produce weathered grains that are log-normally distributed. For the natural patterns, weathering produced lower modality distributions.     

On the coalescence of twisted flux tubes

Abstract

We study the problem of the coalescence of twisted flux tubes by assuming that the azimuthal field lines reconnect at a current sheet during the coalescence process and everywhere else the magnetic field is frozen in the fluid. We derive relations connecting the topology of the coalesced flux tube with the topologies of the initial flux tubes, and then obtain a structure equation for calculating the field configuration of the coalesced flux tube from the given topology. Some solutions for the two extreme cases of low-β plasma and high-β plasma are discussed. The coalesced flux tube has less twist than the initial flux tube. Magnetic helicity is found to be exactly conserved during the coalescence, but the assumptions in the model put a constraint on the energy dissipation so that we do not get a relaxation to the minimum-energy Taylor state in the low-β case. It is pointed out that the structure equation connecting the topology and the equilibrium configuration is quite general and can be of use in many two-dimensional flux tube problems.     

The geochemistry of oceanic basalts in the vicinity of transform faults: Observations and implications

Consideration of geochemical data from basalts formed near major Atlantic and Pacific transform faults reveals two significant sets of observations. First, compared to basalts formed far from the transform, basalts near the ridge/transform intersection have, for the same MgO contents, higher abundances of TiO₂ and other incompatible elements, higher La/Sm and La/Yb ratios, and often higher FeO. These enrichments are distinct from and occur in addition to the more variable and fractionated compositions which have been frequently noted [10–13]. Modeling of this “transform fault effect” using data from the Tamayo/EPR intersection suggests the chemical systematics are caused by decreasing extents of melting as the transform is approached.Second, there are chemical discontinuities in the major element, trace element and isotopic chemistry of basalts across many transforms. These “transform discontinuities” occur in normal ocean crust as well as around hot spots.Consideration of the melting zone in the mantle suggests that the transform fault effect is a natural consequence of the ridge/transform plate boundary. The melting zone beneath a ridge segment must terminate across the transform, leading to lower extents of melting at the transform edge. The surface manifestation of the change in the melting zone may be affected by the age of the transform offset, the spreading rate, the transform spacing and the interaction of mantle flow with the local thermal structure; it may be obscured by episodic magma chamber processes and mantle heterogeneity.The significance of transform discontinuities depends on whether they persist with age. If they do not, then temporally variable crust-forming processes may produce changes along a flow line similar to those at zero age across a transform. If, on the other hand, a discontinuity persists with age, then the transform may be related to a fundamental discontinuity in the underlying mantle. Long-lived transform discontinuities would have profound implications for the nature of plate motions, mantle convection and mantle heterogeneity.     

Magnetohydrodynamic simulation of a solar flare: 2. Flare model and simulation using active-region magnetic maps

The results of a three-dimensional MHD simulation and data obtained using specialized spacecraft made it possible to construct an electrodynamic model of solar flares. A flare results from explosive magnetic reconnection in a current sheet above an active region, and electrons accelerated in field-aligned currents cause hard X rays on the solar surface. In this review, we considered works where the boundary and initial conditions on the photosphere were specified directly from the magnetic maps, obtained by SOHO MDI in the preflare state, in order to simulate the formation of a current sheet. A numerical solution of the complete set of MHD equations, performed using the new-generation PERESVET program, demonstrated the formation of several current sheets before a series of flares. A comparison of the observed relativistic proton spectra and the simulated proton acceleration along a magnetic field singular line made it possible to estimate the magnetic reconnection rate during a flare (∼10⁷ cm s⁻¹). Great flares (of the X class) originate after an increase in the active region magnetic flux up to 10²² Mx.     

Some sources of plasma inhomogeneities in the solar wind in front of the Earth’s magnetosphere

The origination of various plasma inhomogeneities in the magnetosheath in front of the Earth’s magnetosphere is analyzed within classical magnetohydrodynamics. The effect of directional discontinuities or tangential and rotational discontinuities of the solar wind on plasma is studied. The origination of inhomogeneities of the type of secondary MHD waves in the magnetosheath is shown; the former equalize plasma parameters when restoring the stationary state. The effect of a rotational discontinuity on the bow shock–Earth’s magnetosphere system is of special interest, with distinguishing of plasma inhomogeneities of the plateau type observed in the near-Earth space.     

磁岛合并研究

本文采用二维全粒子模拟来研究无碰撞等离子体中的磁岛合并过程.结果表明,磁岛合并分为两个阶段,在第一个阶段,两个磁岛因同向电流丝之间的吸引力而缓慢地相互靠近,在这个过程中,合并线附近的电子被面外电场加速,形成薄电流片,同时电流片两侧形成磁场堆积.第二个阶段为快速重联阶段,合并线附近的电磁场结构和以Harris电流片为初态的重联扩散区的电磁场结构很相似,其中最显著的特点为面外磁场的四极型结构.