The Cartesian method for solving partial differential equations in spherical geometry

Cartesian coordinates are used to solve the nonlinear shallow-water equations on the sphere. The two-dimensional equations, in spherical coordinates, are first embedded in a three-dimensional system in a manner that preserves solutions of the two-dimensional system. That is, solutions of the three-dimensional system, with appropriate initial conditions, also solve the two-dimensional system on the surface of the sphere. The higher dimensional system is then transformed to Cartesian coordinates. Computations are limited to the surface of the sphere by projecting the equations, gradients, and solution onto the surface. The projected gradients are approximated by a weighted sum of function values on a neighboring stencil. The weights are determined by collocation using the spherical harmonics in trivariate polynomial form. That is, the weights are computed from the requirement that the projected gradients are near exact for a small set of spherical harmonics. The method is applicable to any distribution of points and two test cases are implemented on an icosahedral geodesic grid. The method is both vectorizable and parallelizable.     

Rank of coal beds of the Narragansett basin,Massachusetts and Rhode Island

Coal of the Narragansett basin generally has been considered to be anthracite and/or meta-anthracite. However, no single reliable method has been used to distinguish these two ranks in this basin. Three methods — chemical, X-ray, and petrographic — have been used with some degree of success on coal of the Narragansett basin, but too often the results are in conflict. Chemical methods have been limited by inadequate sampling on a coal-bed-by-coal-bed basis and by a lack of analyses made according to (American Society for Testing and Materials, 1974) standard specifications.In addition, when corrections are made by using the Parr formulas, as required by the ASTM (1974) procedures, the generally high to very high ash content of coal from the Narragansett basin causes the fixed-carbon content to appear higher than it actually is. X-ray methods using the degree of graphitization as a measure of rank are not reliable because some of the graphite is related to shearing and brecciation associated with folding and faulting. Petrographic methods using reflectance on vitrinite give results that are generally consistent with results from chemical determinations. However, it is not clear whether the mean maximum reflectance or mean bireflectance is a better indicator of similar rank of such high-rank coals that have been structurally deformed.Coal from the Cranston Mine, RI, is probably meta-anthracite and coal from the Portsmouth Mine is probably anthracite. These ranks are based on chemical,X-ray, and petrographic data and are supported by associated metamorphic mineral assemblages that indicate that the Cranston Mine is in a higher metamorphic zone than the zone containing the Porthmouth Mine. Interpretation of the rank of Mansfield, MA, coal on the basis of extant chemical data is difficult because it is an impure coal with an ash content of 33 to 50%. Reflectance data indicate that the Mansfield, Foxborough, and Plainville coals in the northern part of the Narragansett basin are meta-anthracite but this is in disagreement with the rank suggested by the low degree of metamorphism of the associated rocks.     

A study of conditions at the inner core boundary of the earth

A study is made of the thermal and compositional conditions which the liquid outer core must satisfy at the inner core boundary, assuming the inner core to be growing by continual solidification of the heavy component of the liquid alloy in the outer core. It is found that the outer core is strongly destabilized by the compositional gradients driven by the separation process associated with the freezing. Further, it is argued that all the freezing necessary for the growth of the solid inner core cannot occur on a flat interface; most of it must occur above the solid boundary in a region labeled the slurry layer.     

Rb-Sr whole-rock age of the contact aureole of the Stillwater Igneous Complex,Montana

Fourteen whole-rock samples from three traverses through the contact aureole of the Stillwater Complex were analyzed for Rb, Sr and⁸⁷Sr/⁸⁶Sr. Twelve of these samples yielded an age of 2750 ± 45m.y.; (⁸⁷Sr/⁸⁶Sr)₀ = 0.705 ± 0.003 (2σ). In addition one whole rock and a biotite separate derived from it gave an age of 2544 m.y.; (⁸⁷Sr/⁸⁶Sr)₀ = 0.714. These data support a minimum age of 2750 m.y. for the intrusion of the complex.     

Present-day strain rates at the northern end of the ethiopian rift valley

Five successive geodimeter surveys during a five-year period yield hints as to the manner and rate of crustal extension in the Ethiopian rift valley. The northern geodimeter network traverses an en-echelon offset in the Wonji fault belt at latitude 8°30′N; this belt comprises the youngest volcanism and faulting of the rift floor. The northern network surveys reveal progressive rift extension at mean rates of 3–5 mm/year and strain rates of 6–16 · 10⁻⁷/yr, essentially confined within the Wonji fault belt segments. Small longitudinal motions of persistent dextral sense have occurred in the intervening zone between the offset segments. It is too early to say whether these deformations are local, regional or plate-tectonic phenomena, but the present aseismicity of the rift suggests the buildup of regional strain.     

On the role of the Asian monsoon in the angular momentum and kinetic energy balances of the tropics

The balance conditions of relative angular momentum and time-mean kinetic energy and their annual variations are studied for the Northern Hemisphere tropical belt. The belt is divided into two roughly equal size parts, the monsoon and the extramonsoon regions. The data used consist of all available daily rawinsonde reports from the world areological network for the two 5-year periods 1958–63 and 1968–73.In winter, the trade winds in the monsoon and extramonsoon regions are both sources of westerly relative angular momentum for the middle latitude circulation. However, it is found that the angular momentum gained in the extramonsoon region of the Tropics is mostly destroyed by a net southward flow of mass in that region, and becomes regenerated in the monsoon region by a net northward flow of mass there. This excess of angular momentum together with the angular momentum picked up locally in the monsoon region is almost all exported across its northern boundary. It is further found that in winter the Tropics are also an important source of mean kinetic energy for middle latitudes. Again almost all export of kinetic energy was found to take place across the northern boundary of the monsoon sector. Most of this energy must be generated through the pressure gradient term inside the monsoon region itself, the transformation from transient eddy kinetic energy being very small. The proper evaluation of the pressure gradient appears to be the main stumbling block in the present study, preventing us from estimating the generation and thereby, as a residual, the frictional dissipation in the two regions.In summer, the extramonsoon region remains a source of angular momentum, but the monsoon region with its surface westerlies acts as a sink, leading to a sharp reduction (and even a midsummer reversal) of the export into middle latitudes. Also the export of mean kinetic energy almost vanishes in summer, except for a small southward transfer across the equator. The calculations for two 5-year periods give very similar estimates and thereby show the reliability of the results.Parts of this paper were presented at the International Symposium on Monsoons, March 7–12, 1977 in New Delhi, India.     

A generalization of the Kolmogorov-von Kármán relationship and some further implications on the values of the constants

A general relation between the Kolmogorov (₁) and von Kármán (k) constants appropriate to flows in the surface boundary layer is obtained by including stability and divergence dependent terms in the development. The result is used to show that the author's earlier inference of the valuek = 0.36 drawn from the observed value ,k ^4/3=0.141 cannot have been underestimated due to the neglect of stability or divergence effects. It is concluded that if k is to be 0.4, then ₁ must be 0.5.Work performed under the auspices of the U.S. Energy Research and Development Administration.     

Complex formation in the ternary system Mg(II)-CO2-H2O

The carbonato and hydrogencarbonato complexes of Mg²⁺ were investigated at 25 and 50° in solutions of the constant ClO₄^? molality (3 M) consisting preponderantly of NaClO₄. The experimental data could be explained assuming the following equilibria: Mg²⁺ + CO_2B + H₂O ag MgHCO⁺₃ + H⁺, $\text{log}{}^1\text{β}{}_1\text{= ?7.64}{}_4\text{± 0.01}{}_7\text{(25°), ?7.46}{}_2\text{± 0.01}{}_1\text{(50°)}$, Mg²⁺ + 2 CO_2g + 2 H₂Oag Mg(HCO₃)⁰₂ ± 2 H⁺, $\text{log}{}^1\text{β}{}_2\text{= ?15.00 ± 0.14 (25°)}$, ?15.37 ± 0.39 (50°), Mg²⁺ + CO_2g + H₂Oag MgCO⁰₃ + 2 H⁺, $\text{log}{}^1\text{k}{}_1\text{= ?15.64 ± 0.06 (25°)}$,?15.23 ± 0.02 (50°), with the assumption γ_MgCO3⁰ = γ_Mg(HCO3)⁰2, ΔG⁰(I = 0) for the reaction MgCO⁰₃ + CO_2g + H₂O = Mg(HCO₃)⁰₂ was estimated to be ?3.9₁ ± 0.8₆ and 0.6 ± 2.4 kJ/mol at 25 and 50°C, respectively. The abundance of carbonate linked Mg(II) species in fresh water systems is discussed.