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.     

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.     

Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. VI. Topology of current sheets

Abstract

The electric surface current in a tangential discontinuity in a force-free magnetic field is conserved. The direction of the current is halfway between the direction of the continuous fields on either side of the surface of discontinuity. Hence the current sheets, i.e. the surface of tangential discontinuity, have a topology that is distinct from the lines of force of the field. The precise nature of the topology of the current sheet depends upon the form of the winding patterns in the field. Hence, invariant winding patterns and random winding patterns are treated separately. Current sheets may have edges, at the junction of two or more topological separatrices. The current lines may, in special cases, be closed on themselves. The lines of force that lie on either side of a current sheet somewhere pass off the sheet across a junction onto another sheet. In most cases the current sheets extending along a field make an irregular honeycomb.

The honeycomb pattern varies along the field if the winding pattern of the field varies. The surface current density in a tangential discontinuity declines inversely, or faster, with distance from its region of origin. The edges of weaker tangential discontinuities (originating in more distant regions) are bounded by the stronger tangential discontinuities (of nearby origin).

An examination of the force-free field equations in a small neighborhood of the line of intersection of two tangential discontinuities shows that the lines of force twist around to cross the line of intersection at right angles. If the angle between the tangential discontinuities exceeds π/2, there is also the possibilitity that the lines twist around so as to come tangent to the line of intersection as they cross it.     

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.     

The correlation between gravitational and geomagnetic fields caused by interaction of the core fluid motion with a bumpy core-mantle interface

The correlation discovered by Hide and Malin between the variable parts of the Earth's gravitational field and magnetic field (suitably displaced in longitude) was tentatively and qualitatively explained by them in terms of the influence on both fields of irregularities (or “surface bumps”) at the core-mantle interface. In this paper, a quantitative analysis of this phenomenon is developed, through study of an idealised problem in which conducting fluid occupying the region z < η(x) flows over the surface z = η(x) in the presence of a magnetic field (B₀,0,0), the whole system rotating with angular velocity (0,0,Ω). It is assumed that $\text{|η′(x)| « 1}$ so that perturbation methods are applicable. Determination of the magnetic potential in the “mantle” region z < η(x) requires solution of the full hydromagnetic problem in the fluid. It is shown that three wave modes are excited, two of which (for values of the parameters of the problem of geophysical interest) have a boundary layer character. Phase interactions between these modes lead to a shift and a distortion of the magnetic pattern relative to the gravitational pattern. The correlation between the gravitational potential and the magnetic potential (shifted by a distance x₀) is determined on the plane $\text{z = d (d}\text{a}\text{?}\text{|η|)}$ as a function of x₀/d and the curves obtained are qualitatively similar to that based on the observed data; the maximum correlation obtained varies between 0.67 and 1, depending on values of the parameters of the problem, and is about 0.72 for reasonable estimates of these parameters in the geophysical context.     

Calculations of high-pressure phase transitions in the system MgOSiO2 and implications for mantle discontinuities

By incorporating the knowledge of the observed high-pressure phase transformations, the measured equilibrium phase boundaries for certain transitions, and the measured thermochemical data for certain phases in the system MgOSiO₂, the equilibrium phase boundaries for all the phase transitions in this system at 1000°C have been calculated on the basis of an assumption that volume change across a phase boundary is independent of temperature. Three major seismic velocity discontinuities in the mantle (420, 570, and 650 km) have been chosen for comparison with the measured and calculated equilibrium phase boundaries for the high-pressure phases observed in the system MgOSiO₂. Complications of phase changes due to the addition of FeO and Al₂O₃ to the system have also been accounted for. It is suggested by the results of this study that the 420- and 570-km discontinuities are probably related to phase transitions observed in the system MgOSiO₂, but that the 650-km discontinuity is not likely to be associated with any of the equilibrium phase boundaries observed in olivine, pyroxene, and garnet, but may instead be a chemical change.     

The use of low altitude satellite data bases for modeling of core and crustal fields and the separation of external and internal fields

The near-Earth geomagnetic field is a complex combination of fields from a variety of sources within and external to the Earth. These fields have a complex characterization in space and time such that if the field were known at all times and positions external to the Earth's surface it would be possible to separate the measured field into its component parts except for the low degree field from the crust and the high degree field from the core. In practice the measurements come far short of such a space-time characterization. Because of this it is useful to keep in mind several principles when working with these data. These principles are: (1) know the enemy, i.e., understand as much as possible about all the sources, (2) don't expect a boy to do a man's job, i.e., recognize the limitations of the data, (3) use some common sense, and, (4) recognize personal prejudice. Examples are given from published and unpublished results to illustrate how these principles have been, and ought to be, applied. At the same time, suggestions are made as to how future analyses might proceed. These examples are drawn first of all from the way the main field and near-Earth magnetospheric fields are modeled, then from analyses of residuals from such models resulting from ionospheric and crustal fields.     

An improvement for calculating the correlation between the geomagnetic and the Earth's gravitational fields

There is a correlation between the geomagnetic and the Earth's gravitational fields to a certain degree. We should consider the zeral harmonic terms to calculate the correlation function. Correlation between the geomagnetic non-dipole field of epoch 1980, which was moved eastwards 165; and the departure part of the Earth's gravitational field from the hydrostatic equilibrium, yields a value of 0.6278. It is significant at the 0.01 level. Calculating the correlations of different epochs, we find that the westward drifting velocity of the geomagnetic non-dipole field is 0.3° year⁻¹. It is not necessary to calculate the correlations between any individual harmonic term from either field.     

地壳受力的向量场及奇点的存在性

在地壳的每一点,由于受力定义了整个地壳上的向量场,然后利用同胚,将其转化为球面上的自映射,并根据球面的欧拉示性数等于2,说明该自映射至少有一个不动点,即向量场至少有一个奇点.这样,从理论上解释了地震、海啸、火山等自然现象发生的必然性.     

Hamilton's principle and the vorticity laws for a relativistic perfect fluid

Abstract

The equations for a relativistic perfect fluid result from the requirement that the total mass-energy be stationary with respect to variations δx^α(a, b, c, s) in the space-time location of the fluid particle identified by Lagrangian labels (a, b, c) at the point s on its world-line. By considering variations of the Lagrangian labels that leave the specific volume and entropy unchanged, we obtain a general covariant statement of vorticity conservation. The conservation laws for circulation, potential vorticity, and helicity are simple corollaries. This Noether-theorem derivation shows that the vorticity laws have no analogues in particle mechanics, where the corresponding particle labels cannot be continuously vaned.     

Cauchy integral analogues for the separation and continuation of electromagnetic fields within conducting matter

The main results in the theory of the interpretation of geopotential fields are generalized to the case of arbitrary variable electromagnetic fields by means of elaborating electrodynamic analogues for the integral of the Cauchy type.The generalized Kertz method for separating a variable electromagnetic field into parts related to the sources located in different regions of space is elaborated on the basis of this technique. The generalized Kertz method allows the selection of external and internal, normal and anomalous parts of the geomagnetic field, as well as the separation of geomagnetic anomalies into the surface and deep components caused by conductivity inhomogeneities in the Earth's crust and upper mantle.The theory of analytical continuation of variable electromagnetic fields in a conducting medium is also developed in the present work using the technique of analogues for the integral of the Cauchy type. It is shown that analytical continuation of a field downwards permits the determination of the location and form of deep geoelectric inhomogeneities according to the configuration of the isolines of flux functions for magnetic and electric fields.     

A new airborne Polar Nephelometer for the measurement of optical and microphysical cloud properties. Part II: Preliminary tests

A new optical sensor, the airborne Polar Nephelometer, has been tested in an open wind tunnel. The wind tunnel was operated in cloudy conditions including either cloud water droplets or ice crystals, or a mixture of these particles. The sensor is designed to measure the optical and microphysical parameters of cloud particles sized from a few micrometers to about 500 m diameter. Basically, the probe measures the scattering phase function of an ensemble of cloud particles which intersect a collimated laser beam near the focal point of a paraboloidal mirror. From the measured scattering phase function the retrieval of the droplet-size spectra and subsequent derived quantities such as liquid water content and size parameters can be calculated using an inversion method. The particle phase discrimination (water droplets/ice particles) can be derived from the shape of the scattering phase function and the sensitivity of the probe allows the detection of small ice crystals (typically of 5 m diameter). The paper describes the preliminary results obtained by the prototype version of the Polar Nephelometer in various cloudy conditions. These results are compared with direct microphysical measurements obtained by usual PMS probes also mounted in the wind tunnel. Complementary results obtained in a cold chamber are presented in order to illustrate the reliability of the Polar Nephelometer in the presence of small ice crystals.     

A new airborne polar Nephelometer for the measurements of optical and microphysical cloud properties. Part I: Theoretical design

A new optical sensor, the airborne Polar Nephelometer, is described. The sensor is designed to measure the optical and microphysical parameters of clouds containing either water droplets or ice crystals or a mixture of these particles ranging in size from a few micrometers to about 500 m diameter. The probe measures the scattering phase function of an ensemble of cloud particles intersecting a collimated laser beam near the focal point of a paraboloïdal mirror. The light scattered from polar angles from 3.49° to 169° is reflected onto a circular array of 33 photodiodes. The signal processing electronics and computer storage can provide one measurement of the scattering phase function every 100 ms or every 0.2 ms. The first part of the paper describes the theoretical design of a prototype version of the probe.     

High resolution profiles for arsenic in the Seine Estuary. Seasonal variations and net fluxes to the English Channel

Three sampling cruises were conducted in the Seine Estuary from 1993 to 1995 in varying hydrological and seasonal conditions. The site included all of the lower part of the river under the influence of tidal dynamics and the dilution plume in the Baie de Seine. Chemical speciation of arsenic showed high seasonal variations, especially in September when AsIII represented around 50% of dissolved arsenic. The inclusion of organoarsenic compounds not accessible to direct analysis by hydride generation required preliminary mineralisation of the samples. The ratio of dissolved to particulate arsenic distribution was controlled mainly by the iron content of particles. Biological activity had an influence on chemical speciation and thus on the partition coefficient (K_D 10⁻³=6±1 in September and 12±0.9 in February). The zone of conservative mixing used for Seine River flow calculations was limited to a salinity range of approximately 10–30. Dissolved arsenic concentrations extrapolated to null salinities were lower during the high-water period because of dilution (17.6±1.1 nM), and maximal during the low-water period in summer (35.7±0.9 nM). Mean arsenic export to the English Channel was estimated at 33.2±6 T yr⁻¹ for dissolved arsenic. Observation of an arsenic output greater than the upstream input, as well as a simultaneous increase in dissolved and particulate arsenic concentrations during the mixing of freshwater with seawater, strongly suggested the existence of an important intra-estuarine source of arsenic, either of industrial origin or related to the transport and diagenesis of marine sediments.     

Dynamics of equilibrium upset and electromagnetic energy transformation in the geomagnetotail: A theory and simulation using particles. 3. Versions of formation of thin current sheets

The final part of the study is devoted to the process of rapid reconfiguration of the geomagnetotail after the upset of the equilibrium, i.e., to substorm activation. Such a version of spontaneous formation of nonlinear kinetic small-scale structures at the site occupied by an initial current sheet (CS), which results in the formation of a specific equilibrium current structure with strong ion anisotropy (forced kinetic current sheet, FKCS), has been revealed in the course of numerical simulation. This reconfiguration “channel” is realized when the relative value of the magnetic field component normal to CS is smaller than a certain critical value. In this case the disturbance intensity is SPONTANEOUSLY settled at a certain final level at the late stages. The obtained results are compared with the previous study of the system evolution under the action of the external trigger. The physical causes and specific features of different CS evolution regimes are considered. The studied plasma mechanisms are responsible for the effects of rapid energy transformation—magnetic field “annihilation”—that took place in the geomagnetotail.     

Training the rivers and exploring the coasts. Knowledge evolution in the Netherlands in two engineering fields between 1800 and 1940

In this paper, the author wants to explore the knowledge development in two crucial fields, river management and coast management in the 19th century and first decades of the 20th century. Were there similar characteristics in this development? Which types of knowledge can be distinguished? Who were the principal actors in these processes? Did the knowledge evolution have a Dutch stamp or a rather international flavour?To structure the analysis, the author uses the concept of technology regime, a set of technical rules which shapes the know-how of engineers, their design rules and research processes. The analysis shows that the knowledge development of river management and coastal management followed different evolution paths between 1800 and 1940. In the field of river management, a substantial amount of mathematical and physical theories had been gradually developed since the end of the 17th century. After 1850, the regularization approach met gradually a widespread support. Empirical data, design rules, theoretical knowledge and engineering pivoted around the regularization approach, and a technology regime around this approach emerged. The regularization regime further developed in the 20th century, and handbooks were increasingly shaped by mathematical and physical reasoning and formulas. On the other hand, coastal management was until the 1880s a rather marginal activity. Coastal engineering was an extremely complex and multidimensional field of knowledge which no engineer was able to grasp.The foundation of a Dutch weather institute was a first important step towards a more theoretical approach. The Zuiderzee works (starting in 1925) gave probably the most important stimuli to scientific coastal research. It was also a main factor in setting up scientific institutes by Rijkswaterstaat. So from the 1920s, Rijkswaterstaat became a major producer of scientific knowledge, not only in tidal modelling but also in coastal research. Due to a multidisciplinary knowledge network, coastal research transformed from a marginal to a first-rank scientific field, and this transformation enabled Rijkswaterstaat to set a much higher level of ambition in coastal management. The 1953 flood and the Deltaworks marked a new era. New design rules for sea dykes and river levees, based on a revolutionary statistical risk approach were determined, and design rules for the Deltaworks estuary closures were developed, being enabled by the development of hydraulic research.