An analytic model of the Earth's magnetosphere

An analytic magnetic field model for the Earth's magnetosphere is constructed from a dipole field and a tail field. This model can be taken as a generalization of the Dungey's model, after one adds to it a horizontal component. The magnetic topology in the noon-midnight meridian plane of this model is fully determined and it is compared with the topology of other models. In this study it is found that, for a specific value of the parameterk, which is associated to any form of the model, the noon's side neutral points obey a bifurcation scheme.     

A mathematical model of a cloud

The model under consideration is a pencil of radiation incident on a cloud, and the problem is to determine the reflection and transmitted radiation. Based on the method of principle of invariance, three mathematical models are constructed. The first is the basic model, which describes the radiation system completely. The second is the flux integral model, in which the integral average intensity is considered. The third is the diffusion model, which gives the most important information about the diffused radiation field.     

State studies of Earth's plasmasphere: A review

The plasmasphere sandwiched between the ionosphere and the outer magnetosphere is populated by up flow of ionospheric cold (∼1 eV) and dense plasma along geomagnetic field lines. Recent observations from various instruments onboard IMAGE and CLUSTER spacecrafts have made significant advances in our understanding of plasma density irregularities, plume formation, erosion and refilling of the plasmasphere, presence of thermal structures in the plasmasphere and existence of radiation belts. Still modeling work and more observational data are required for clear understanding of plasmapause formation, existence of various sizes and shapes of density structures inside the plasmasphere as well as on the surface of the plasmapause, plasmasphere filling and erosion processes; which are important in understanding the relation of the process proceeding in the Sun and solar wind to the processes observed in the Earth's atmosphere and ionosphere.     

A mathematical magnetospheric field model with independent physical parameters

A quantitative magnetospheric magnetic field model has been calculated in three dimensions. The model is based on an analytical solution of the Chapman-Ferraro problem. For this solution, the magnetopause was assumed to be an infinitesimally thin discontinuity with given geometry. The shape of the dayside magnetopause is in agreement with measurements derived from spacecraft boundary crossings.The magnetic field of the magnetopause currents can be derived from scalar potentials. The scalar potentials result from solutions of Laplace's equation with Neumann's boundary conditions. The boundary values and the magnetic flux through the magnetopause are determined by all magnetic sources which are located inside and outside the magnetospheric cavity. They include the Earth's dipole field, the fields of the equatorial ring current and tail current systems, and the homogeneous interplanetary magnetic field. In addition, the flux through the magnetopause depends on two constants of interconnection which provide the possibility of calculating static interconnection between magnetospheric and interplanetary field lines. Realistic numerical values for both constants have been derived empirically from observed displacements of the polar cusps which are due to changes in the orientation of the interplanetary field. The transition from a closed to an open magnetosphere and vice versa can be computed in terms of a change of the magnetic boundary conditions on the magnetopause. The magnetic field configuration of the closed magnetosphere is independent of the amount and orientation of the interplanetary field. In contrast, the configuration of the open magnetosphere confirms the observational finding that field line interconnection occurs primarily in the polar cusp and high latitude tail regions.The tail current system reflects explicitly the effect of dayside magnetospheric compression which is caused by the solar wind. In addition, the position of the plasma sheet relative to the ecliptic plane depends explicitly on the tilt angle of the Earth's dipole. Near the tail axis, the tail field is approximately in a self-consistent equilibrium with the tail currents and the isotropic thermal plasma.The models for the equatorial ring current depend on the D_st-parameter. They are self-consistent with respect to measured energy distributions of ring current protons and the axially symmetric part of the magnetospheric field.     

A tentative mathematical description of the oscillating peak model of multi-ring basin formation

The oscillating peak model of basin and crater formation proposed by Murray is analysed mathematically. The results could be compared with some basins in order to confirm the basic ideas of this model.     

A mathematical model of the initial stage in the formation of a disk galaxy

The collapse of a homogeneous, initially spheroidal halo under self gravitation, has been considered. It is found that a weak magnetic field (as is plausible to belong to such a cloud) has little influence on the collapse, except probably sufficiently close to the centre where the gas density, and consequently the magnetic field, becomes rather high. The equatorial collapse is centrifugally balanced at a certain stage, while collapse in the perpendicular direction continues. A thick stellar disk is formed within a time-scale <3×10⁹ yr. Brisk star formation takes place while the collapse of the gaseous disk is still in progress. This gives rise to the halo stars with low metal content and high Z-motion. A bulge is formed at the centre simultaneously. This is the first phase of formation of a disk galaxy. The thin disk is formed at a later stage as the remaining primordial gas and the gas released by the evolution of stars in the thick disk gradually settles on to it.The presented model is rather a crude one. Many aspects have not been considered, and many details have not been worked out. It is hoped that a more detailed and comprehensive model will be arrived at in the future.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

A sensitivity study of changes in Earth's rotation rate with an atmospheric general circulation model

A series of general circulation model simulations in which Earth's rotation rate has been increased is presented using the community climate model version 1 (CCM1) of the National Center for Atmospheric Research (NCAR). The rotation rate has been altered in order to simulate day-lengths of 24-, 22-, 20-, 18-, 16- and 14-h Earth days. This is a plausible range of Earth's day-length during the last 4 billion years. In an earlier study with a simple energy balance ocean, which does not store heat (sometimes referred to as a swamp), it was shown that reducing the day-length to a 14-h day caused a 20% reduction in the global mean cloud fraction. In this study however, using fixed sea surface temperatures (SSTs) with January solar forcing, a slight increase in clouds occurs with faster rotation, although changes in relative humidity are similar to the earlier study. Furthermore, as in an earlier study, there is more sinking in the mid-latitudes. This sinking is most prominent over the Pacific and Atlantic storm tracks, indicating that the baroclinic eddies have been weakened. With faster rotation rates, the storm tracks are defined by shorter waves as compared to the control simulation. A significant change in the large-scale zonally averaged circulation occurs when the day-length is reduced to a values less than 18-hours. The effects of faster rotation rates on stationary eddy heat transport may help to explain high latitude glaciation of the Ordovician some 440 Ma years ago.     

A new investigation of the secular drift of the Earth's pole

A new investigation of the secular drift of the Earth's pole was made based on nine long series of latitude observations. This led to the following conclusions: 1) During this century, the Earth's pole has been moving approximately along meridian 70 °W at a mean rate of 0.0016″/yr, much less than the 0.0035″/yr derived from the ILS sequence. 2) Relative to the North American Continent, the Ukiah Station located on the western shore of the U.S. shows a local drift of about 6 cm/yr toward north, in good agreement with the result from the new techniques. 3) Relative to the Europe-Asia plate, the whole North American Continent shows a northward drift of about 8 cm/yr. The Mediterranean shows a similar drift of about 6 cm/yr. 4) Three of five ILS stations, Ukiah, Gaithersberg and Carloforte, show significant drifts. Therefore, the Conventional International Origin (CIO), which is defined by the mean latitudes at 1903.0 of the five stations, is far from being fixed on the Earth's surface, and is not suitable as the origin of the Earth Reference System.     

A relation between solar activity and the Earth's rotation

In this paper, applying Vondrák band filter to both series of (l.o.d.) and sunspot relative number (R), we obtain variations of amplitude of 11 yr term during 1800–1985. The results show that solar cyclic signal in (l.o.d.) series is weak and unstable. The amplitude of 11 yr term in R series has long-periodic variation. The paper has briefly discussed some results about effects of solar activity on the Earth's rotation through the atmospheric motion. From the variation of (l.o.d.) obtained by band filter, we find that maxima of amplitude of annual term in (l.o.d.) occur at the same time with those of sunspot number. It implies that the angular momentum imbalance between the circulations in Southern Hemisphere and Northern Hemisphere is controlled in some way by solar activity.     

Viscosity of the lithosphere of Enceladus

High resolution Voyager II images of Enceladus reveal that some regions on its surface are highly cratered; the most heavily cratered surfaces probably date back to a period of heavy bombardment. The forms of many of the craters on Enceladus are similar to those of fresh lunar craters, but many of the craters are much shallower in depth, and the floors of some craters are bowed up. The flattering of craters and bowing up of the floors are indicative of viscous relaxation of the topography. Analysis of the forms of the flattened craters suggests that the viscosity at the top of the lithosphere, in the most heavily cratered regions, is between 10²⁴ and 10²⁵ P. The exact time scale for the collapse of the craters is not known, but probably was between 100 my and 4 gy. The flattened craters are located in distinct zones that are adjacent to zones, of similar age, where craters have not flattened. The zones where flattened craters occur possibly are regions in which the heat flow was (or is) higher than that in the adjacent terrains. Because the temperature at the top of the lithosphere of Enceladus would be less than or equal to that of Ganymede and Callisto, if it is covered by a thick regolith, and because the required viscosity on Enceladus is one to two orders of magnitude less than that for Ganymede and Callisto, it can be concluded that the lithospheric material on Enceladus is different from that of Ganymede and Callisto. Enceladus probably has a mixture of ammonia ice and water ice in the lithosphere, whereas the lithospheres of Ganymede and Callisto are composed primarily of water ice.     

The strength of the lunar lithosphere

A self-gravitating, elastic, spherical thick shell model is used to derive the present state of the lateral variations of density and stress differences within the lunar lithosphere. The model is allowed to deform under the load of an initial surface topography and internal density distribution, such that the resulting deformed body gives rise to the observed surface topography and gravity specified by the spherical harmonics of degree up to 70. Two main models are considered, Model A and Model B, with elastic lithospheres of thickness 300 and 210 km, respectively. Model A displays density perturbations of generally less than ±200 kg/m³ within the crustal layers, reducing rapidly to less than ±20 kg/m³ at the base of the lithosphere. The density perturbations in Model B are similar in the crust and marginally higher at the base of the lithosphere. The major stress differences in the mantle are associated with the mascon basins and are found to reach maximums of 8-10 MPa within the lower lithosphere (150-270 km) of Model A and maximums of 12-16 MPa at 150 to 180 km depth for Model B. A moderate correlation exists between the modeled stress distributions and shallow moonquake epicenters. However, the overall results of this study imply that other remnant stresses, due to processes other than density perturbations, exist and play a critical role in the large shallow moonquakes.     

Strength of the lithosphere of the Galilean satellites

Several approaches have been used to estimate the ice shell thickness on Callisto, Ganymede, and Europa. Here we develop a method for placing a strict lower bound on the thickness of the strong part of the shell (lithosphere) using measurements of topography. The minimal assumptions are that the strength of faults in the brittle lithosphere is controlled by lithostatic pressure according to Byerlee's law and the shell has relatively uniform density and thickness. Under these conditions, the topography of the ice provides a direct measure of the bending moment in the lithosphere. This topographic bending moment must be less than the saturation bending moment of the yield strength envelope derived from Byerlee's law. The model predicts that the topographic amplitude spectrum decreases as the square of the topographic wavelength. This explains why Europa is rugged at shorter wavelengths (∼10 km) but extremely smooth, and perhaps conforming to an equipotential surface, at longer wavelengths (>100 km). Previously compiled data on impact crater depth and diameter [Schenk, P.M., 2002. Nature 417, 419-421] on Europa show good agreement with the spectral decrease predicted by the model and require a lithosphere thicker than 2.5 km. A more realistic model, including a ductile lower lithosphere, requires a thickness greater than 3.5 km. Future measurements of topography in the 10-100 km wavelength band will provide tight constraints on lithospheric strength.     

Representation of the Earth's gravitational potential

The paper represents the Earth's gravitational potentialV, outside a sphere bounding the Earth, by means of its difference V from the author's spheroidal potential. The difference V is in turn represented as arising from a surface density on the sphere bounding the Earth. Because of the slow decrease with ordern of the normalized coefficients in the spherical harmonic expansion ofV, the density anomalies from which the higher coefficients arise must occur in regions close to the Earth's surface. The surface density is thus an idealization of the product of the density anomaly and the crustal thicknessb. Values of are computed from potential coefficients obtained from two sources, Rapp and the Smithsonian Astrophysical Observatory. The two sources give qualitative agreement for the values of and for its contour map. The numerical values obtained for are compatible with the idea that the responsible density anomalies are reasonably small, i.e., less than 0.05 g/cm³, and occur in the crust alone.This paper was prepared under the sponsorship of the Electronics Research Center of the National Aeronautics and Space Administration through NASA Grant NGR 22-009-311.     

Dimension of the Earth's General Ellipsoid

The problem of specifying the Earth's mean (general)ellipsoid is discussed. This problem has been greatly simplified in the era of satellite altimetry, especially thanks to the adopted geoidal geopotential value, W₀ = (62 636 856.0 ± 0.5) m² s^-2.Consequently, the semimajor axis a of the Earth's mean ellipsoid can be easily derived. However, an a priori condition must be posed first. Two such a priori conditions have been examined, namely an ellipsoid with the corresponding geopotential that fits best W₀ in the least squares sense and an ellipsoid that has the global geopotential average equal to W₀. It has been demonstrated that both a priori conditions yield ellipsoids of the same dimension, with a–values that are practically identical to the value corresponding to the Pizzetti theory of the level ellipsoid: a = (6 378 136.68 ± 0.06) m.     

Hard X-ray emission of the Earth's atmosphere: Monte Carlo simulations

We perform Monte Carlo simulations of cosmic ray-induced hard X-ray radiation from the Earth's atmosphere. We find that the shape of the spectrum emergent from the atmosphere in the energy range 25–300 keV is mainly determined by Compton scatterings and photoabsorption, and is almost insensitive to the incident cosmic ray spectrum. We provide a fitting formula for the hard X-ray surface brightness of the atmosphere as would be measured by a satellite-borne instrument, as a function of energy, solar modulation level, geomagnetic cut-off rigidity and zenith angle. A recent measurement by the INTEGRAL observatory of the atmospheric hard X-ray flux during the occultation of the cosmic X-ray background by the Earth agrees with our prediction within 10 per cent. This suggests that Earth observations could be used for in-orbit calibration of future hard X-ray telescopes. We also demonstrate that the hard X-ray spectra generated by cosmic rays in the crusts of the Moon, Mars and Mercury should be significantly different from that emitted by the Earth's atmosphere.     

A mathematical solution for the formation of halo

The behaviour of dense astrophysical systems can be described by the magnetohydrodynamic equations involving gravitational potential. In this paper the magnetohydrodynamic equations are solved in the wave form and a general dispersion relation have been obtained. This dispersion relation has been used with simplifying assumptions, plausible for special regions of the system and results obtained have been shown to be able to interpret the property of the gas in those special regions. For example, a region at a large distance from the centre of the system — i.e.,r — is considered. The analysis indicates that instability exists at such a large distance though it is assumed that the region is homogeneous. This explains the formation of corona, envelope of supergiant or galactic halo.     

A case study of Kelvin-Helmholtz vortices on both flanks of the Earth's magnetotail

Kelvin-Helmholtz instability (KHI) is a fundamental fluid dynamical process that develops in a velocity shear layer. It is excited on the tail-flanks of the Earth's magnetosphere where the flowing magnetosheath plasma and the stagnant magnetospheric plasma sit adjacent to each other. This instability is thought to induce vortical structures and play an important role in plasma transport there. While KHI vortices have been detected, the earlier observations were performed only on one flank at a time and questions related to dawn-dusk asymmetry were not addressed. Here, we report a case where KHI vortices grow more or less simultaneously and symmetrically on both flanks, despite all the factors that may have broken the symmetry. Yet, energy distributions of ions in and around the vortices show a remarkable dawn-dusk asymmetry. Our results thus suggest that although the initiation and development of the KHI depend primarily on the macroscopic properties of the flow, the observed enhancement of ion energy transport around the dawn side vortices may be linked to microphysical processes including wave-particle interactions. Possible coupling between macro- and micro-scales, if it is at work, suggests a role for KHI not only within the Earth's magnetosphere (e.g., magnetopause and geomagnetic tail) but also in other regions where shear flows of magnetized plasma play important roles.     

Aerial electromagnetic sounding of the lithosphere of Venus

Electromagnetic (EM) investigation depths are larger on Venus than Earth due to the dearth of water in rocks, in spite of higher temperatures. Whistlers detected by Venus Express proved that lightning is present, so the Schumann resonances ～10–40 Hz may provide a global source of electromagnetic energy that penetrates ～10–100 km. Electrical conductivity will be sensitive at these depths to temperature structure and hence thermal lithospheric thickness. Using 1D analytic and 2D numerical models, we demonstrate that the Schumann resonances—transverse EM waves in the ground-ionosphere waveguide—remain sensitive at all altitudes to the properties of the boundaries. This is in marked contrast to other EM methods in which sensitivity to the ground falls off sharply with altitude. We develop a 1D analytical model for aerial EM sounding that treats the electrical properties of the subsurface (thermal gradient, water content, and presence of conductive crust) and ionosphere, and the effects of both random errors and biases that can influence the measurements. We initially consider specified 1D lithospheric thicknesses 100–500 km, but we turn to 2D convection models with Newtonian temperature-dependent viscosity to provide representative vertical and lateral temperature variations. We invert for the conductivity-depth structure and then temperature gradient. For a dry Venus, we find that the error on temperature gradient obtained from any single local measurement is ～100%—perhaps enough to distinguish “thick” vs. “thin” lithospheres. When averaging over thousands of kilometers, however, the standard deviation of the recovered thermal gradient is within the natural variability of the convection models, <25%. A “wet” interior (hundreds of ppm H₂O) limits EM sounding depths using the Schumann resonances to <20 km, and errors are too large to estimate lithospheric properties. A 30-km conductive crust has little influence on the dry-interior models because the Schumann penetration depths are significantly larger. We conclude that EM sounding of the interior of Venus is feasible from a 55-km high balloon. Lithospheric thickness can be measured if the upper-mantle water content is low. If H₂O at hundreds of ppm is present, the deeper, temperature-sensitive structure is screened, but the “wet” nature of the upper mantle, as well as structure of the upper crust, is revealed.     

Long-term variations of the Earth's orbital elements

Critical analysis of theories of the long-term variations of the ecliptical elements of the Earth leads to the following conclusions, regarding the influence of different terms on the accuracy of the expansions used:

1. further improvement in planetary masses will not have significant influence:
2. for the (e, π) system, terms depending upon the second order as to the disturbing masses are more important than ones coming from the third degree with respect to the planetary eccentricities and inclinations;
3. for the (i, Ω) system, the latter terms have highly significant influence, whereas additional terms in masses are negligible. The same conclusion can be drawn for (ε,Ψ _g ). Using these results, a new solution for the long-term variations of the Earth's orbital elements is obtained. The results fore, π,i, Ω include terms depending upon the second power as to the disturbing masses and to the third degree with respect to the planetarye's andi's. For the obliquity ε and the annual general precession in longitudeΨ _g , a Laplace series is proposed where amplitudes, mean rates and phases are computed from those of the (i, Ω) system.