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Concept# Earth ellipsoid

Summary

An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.
It is a spheroid (an ellipsoid of revolution) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the equatorial axis (a) and the polar axis (b); their radial difference is slightly more than 21 km, or 0.335% of a (which is not quite 6,400 km).
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924,

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Jiannong Fang, Fernando Porté Agel

In the last few decades, laboratory experiments and direct numerical simulations of turbulent boundary layers, performed at low to moderate Reynolds numbers, have found very-large-scale motions (VLSMs) in the logarithmic and outer regions. The size of VLSMs was found to be 10-20 times as large as the boundary-layer thickness. Recently, few studies based on field experiments examined the presence of VLSMs in neutral atmospheric boundary-layer flows, which are invariably at very high Reynolds numbers. Very large scale structures similar to those observed in laboratory-scale experiments have been found and characterized. However, it is known that field measurements are more challenging than laboratory-based measurements, and can lack resolution and statistical convergence. Such challenges have implications on the robustness of the analysis, which may be further adversely affected by the use of Taylor’s hypothesis to convert time series to spatial data. We use large-eddy simulation (LES) to investigate VLSMs in atmospheric boundary-layer flows. In order to make sure that the largest flow structures are properly resolved, the horizontal domain size is chosen to be much larger than the standard domain size. It is shown that the contributions to the resolved turbulent kinetic energy and shear stress from VLSMs are significant. Therefore, the large computational domain adopted here is essential for the purpose of investigating VLSMs. The spatially coherent structures associated with VLSMs are characterized through flow visualization and statistical analysis. The instantaneous velocity fields in horizontal planes give evidence of streamwise-elongated flow structures of low-speed fluid with negative fluctuation of the streamwise velocity component, and which are flanked on either side by similarly elongated high-speed structures. The pre-multiplied power spectra and two-point correlations indicate that the scales of these streak-like structures are very large. These features are similar to those found in the logarithmic and outer regions of laboratory-scale boundary layers by direct numerical simulation and experiments conducted at low to moderate Reynolds numbers. The three-dimensional correlation map and conditional average of the three components of velocity further indicate that the low-speed and high-speed regions possess the same elongated ellipsoid-like structure, which is inclined upward along the streamwise direction, and they are accompanied by counter-rotating roll modes in the cross-section perpendicular to the streamwise direction. These results are in agreement with recent observations in the atmospheric surface layer. Furthermore, the influences of the Coriolis force, the domain size, and the subgrid-scale model on VLSMs are investigated.

2015,

Large-eddy simulation is used to investigate very-large-scale motions (VLSMs) in the neutrally stratified atmospheric boundary layer at a very high friction Reynolds number, . The vertical height of the computational domain is m, which corresponds to the thickness of the boundary layer. In order to make sure that the largest flow structures are properly resolved, the horizontal domain size is chosen to be and , which is much larger than the standard domain size, especially in the streamwise direction (i.e., the direction of elongation of the flow structures). It is shown that the contributions to the resolved turbulent kinetic energy and the resolved shear stress from streamwise wavelengths larger than are up to 27 and 31 % respectively. Therefore, the large computational domain adopted here is essential for the purpose of investigating VLSMs. The spatially coherent structures associated with VLSMs are characterized through flow visualization and statistical analysis. The instantaneous velocity fields in horizontal planes give evidence of streamwise-elongated flow structures of low-speed fluid with negative fluctuation of the streamwise velocity component, and which are flanked on either side by similarly elongated high-speed structures. The pre-multiplied power spectra and two-point correlations indicate that the scales of these streak-like structures are very large, up to in the streamwise direction and in the spanwise direction. These features are similar to those found in the logarithmic and outer regions of laboratory-scale boundary layers by direct numerical simulation and experiments conducted at low to moderate Reynolds numbers. The three-dimensional correlation map and conditional average of the three components of velocity further indicate that the low-speed and high-speed regions possess the same elongated ellipsoid-like structure, which is inclined upward along the streamwise direction, and they are accompanied by counter-rotating roll modes in the cross-section perpendicular to the streamwise direction. These results are in agreement with recent observations in the atmospheric surface layer.